← Corresponding and alternate angles are equal · Angle Facts in a Figure

Corresponding and alternate angles are equal · 10 practice problems

4.G.A.1

0 / 10 solved

Show all answers (parent / teacher)

Generated variants — 10

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 easy

In the figure, line $p$ and line $q$ are parallel to each other. Find the measure of angle $a$ .

[Figure] Two horizontal parallel lines, $p$ (top) and $q$ (bottom), are crossed by two transversal lines that intersect in an X shape. Where a transversal meets the upper line $p$ , one angle is marked as $40^\circ$ . Where they meet the lower line $q$ , one angle is marked as $40^\circ$ , and the angle $a$ is marked at the crossing point between the lines.

Your answer

Show solution

1 · Understandwhat's really being asked

Two parallel lines p (top) and q (bottom) are crossed by two slanted lines that meet in an X between them. One angle where a transversal meets the top line p is 40 deg; one angle where a transversal meets the bottom line q is 40 deg. At the crossing point between the lines, the angle a that opens downward toward q is marked. I need that angle a.

Givens

Line p and line q are parallel.
Two transversal lines cross between p and q, forming a triangle whose third side lies along q.
An angle made with line p is 40 deg.
A base angle made with line q is 40 deg.
The angle a is the apex angle of the triangle, at the crossing point, opening toward q.

Unknowns

The measure of the angle a at the crossing point.

Constraints

Alternate interior angles between parallel lines are equal.
The three interior angles of a triangle add to 180 deg.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

The two transversals and line q bound a triangle. Its apex is the crossing point (the angle a), and its base sits on q. The 40 deg is one base angle directly. The 40 deg at line p moves down to the other base angle on q by alternate interior angles (p is parallel to q). Then the apex angle is what is left to reach 180 deg.

3 · Execute3 carry out the plan

1Spot the triangle

#1 Draw a Diagram 4.G.A.1

The two slanted lines meet at the crossing point and continue down to line q, cutting off a triangle whose apex is the crossing point and whose base lies on q. The angle a is the apex angle.

a = 180^\circ - (\text{base angle}_1) - (\text{base angle}_2)

Drawing the picture turns 'angles between parallel lines' into a single triangle whose angles must add to 180 deg.

2Move the 40 deg down to line q

#7 Identify Subproblems 4.G.A.2

A transversal crosses the parallel lines p and q. The 40 deg angle it makes with p equals its alternate interior angle with q, so one base angle of the triangle on q is 40 deg.

40^\circ \text{ at } p = 40^\circ \text{ base angle at } q

Because p and q are parallel, a slanted line crosses them at the same tilt, so the matching (alternate) angle is also the same.

3Use the two base angles

#7 Identify Subproblems 4.MD.C.7

The triangle's two base angles on q are 40 deg and 40 deg. The apex angle a is 180 deg minus their sum.

a = 180^\circ - 40^\circ - 40^\circ = 100^\circ

Once two angles of a triangle are known, the third is forced, because all three always total 180 deg.

Answer: 100 degrees

4 · Reviewdoes it hold up?

The base angles 40 deg and 40 deg add to 80 deg, leaving 100 deg for the apex. All three angles 40 + 40 + 100 = 180 deg.

Another way: Use the exterior angle idea (tool 7): the angle the crossing makes on the p-side is the exterior angle of the same triangle and equals 40 + 40 = 80 deg; the angle a is its straight-line partner, 180 - 80 = 100 deg.

Standardsmin grade 4

4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Identifying the triangle bounded by the two transversals and line q.
4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Using the parallel lines to move the 40 deg angle to its alternate interior angle on q.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting 40 and 40 from 180 to get the apex angle.

💡Takeaway. Find the hidden triangle, slide the top angle down between the parallel lines, and the last angle is just what is left of 180 deg!

Variant 2 easy

In the figure, line $p$ and line $q$ are parallel to each other. Find the measure of angle $a$ .

[Figure] Two horizontal parallel lines, $p$ (top) and $q$ (bottom), are crossed by two transversal lines that intersect in an X shape. Where a transversal meets the upper line $p$ , one angle is marked as $45^\circ$ . Where they meet the lower line $q$ , one angle is marked as $35^\circ$ , and the angle $a$ is marked at the crossing point between the lines.

Your answer

Show solution

1 · Understandwhat's really being asked

Two parallel lines p (top) and q (bottom) are crossed by two slanted lines that meet in an X between them. One angle where a transversal meets the top line p is 45 deg; one angle where a transversal meets the bottom line q is 35 deg. At the crossing point between the lines, the angle a that opens downward toward q is marked. I need that angle a.

Givens

Line p and line q are parallel.
Two transversal lines cross between p and q, forming a triangle whose third side lies along q.
An angle made with line p is 45 deg.
A base angle made with line q is 35 deg.
The angle a is the apex angle of the triangle, at the crossing point, opening toward q.

Unknowns

The measure of the angle a at the crossing point.

Constraints

Alternate interior angles between parallel lines are equal.
The three interior angles of a triangle add to 180 deg.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

The two transversals and line q bound a triangle. Its apex is the crossing point (the angle a), and its base sits on q. The 35 deg is one base angle directly. The 45 deg at line p moves down to the other base angle on q by alternate interior angles (p is parallel to q). Then the apex angle is what is left to reach 180 deg.

3 · Execute3 carry out the plan

1Spot the triangle

#1 Draw a Diagram 4.G.A.1

The two slanted lines meet at the crossing point and continue down to line q, cutting off a triangle whose apex is the crossing point and whose base lies on q. The angle a is the apex angle.

a = 180^\circ - (\text{base angle}_1) - (\text{base angle}_2)

Drawing the picture turns 'angles between parallel lines' into a single triangle whose angles must add to 180 deg.

2Move the 45 deg down to line q

#7 Identify Subproblems 4.G.A.2

A transversal crosses the parallel lines p and q. The 45 deg angle it makes with p equals its alternate interior angle with q, so one base angle of the triangle on q is 45 deg.

45^\circ \text{ at } p = 45^\circ \text{ base angle at } q

Because p and q are parallel, a slanted line crosses them at the same tilt, so the matching (alternate) angle is also the same.

3Use the two base angles

#7 Identify Subproblems 4.MD.C.7

The triangle's two base angles on q are 45 deg and 35 deg. The apex angle a is 180 deg minus their sum.

a = 180^\circ - 45^\circ - 35^\circ = 100^\circ

Once two angles of a triangle are known, the third is forced, because all three always total 180 deg.

Answer: 100 degrees

4 · Reviewdoes it hold up?

The base angles 45 deg and 35 deg add to 80 deg, leaving 100 deg for the apex. All three angles 45 + 35 + 100 = 180 deg.

Another way: Use the exterior angle idea (tool 7): the angle the crossing makes on the p-side is the exterior angle of the same triangle and equals 45 + 35 = 80 deg; the angle a is its straight-line partner, 180 - 80 = 100 deg.

Standardsmin grade 4

4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Identifying the triangle bounded by the two transversals and line q.
4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Using the parallel lines to move the 45 deg angle to its alternate interior angle on q.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting 45 and 35 from 180 to get the apex angle.

💡Takeaway. Find the hidden triangle, slide the top angle down between the parallel lines, and the last angle is just what is left of 180 deg!

Variant 3 easy

In the figure, line $p$ and line $q$ are parallel to each other. Find the measure of angle $a$ .

[Figure] Two horizontal parallel lines, $p$ (top) and $q$ (bottom), are crossed by two transversal lines that intersect in an X shape. Where a transversal meets the upper line $p$ , one angle is marked as $55^\circ$ . Where they meet the lower line $q$ , one angle is marked as $40^\circ$ , and the angle $a$ is marked at the crossing point between the lines.

Your answer

Show solution

1 · Understandwhat's really being asked

Two parallel lines p (top) and q (bottom) are crossed by two slanted lines that meet in an X between them. One angle where a transversal meets the top line p is 55 deg; one angle where a transversal meets the bottom line q is 40 deg. At the crossing point between the lines, the angle a that opens downward toward q is marked. I need that angle a.

Givens

Line p and line q are parallel.
Two transversal lines cross between p and q, forming a triangle whose third side lies along q.
An angle made with line p is 55 deg.
A base angle made with line q is 40 deg.
The angle a is the apex angle of the triangle, at the crossing point, opening toward q.

Unknowns

The measure of the angle a at the crossing point.

Constraints

Alternate interior angles between parallel lines are equal.
The three interior angles of a triangle add to 180 deg.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

The two transversals and line q bound a triangle. Its apex is the crossing point (the angle a), and its base sits on q. The 40 deg is one base angle directly. The 55 deg at line p moves down to the other base angle on q by alternate interior angles (p is parallel to q). Then the apex angle is what is left to reach 180 deg.

3 · Execute3 carry out the plan

1Spot the triangle

#1 Draw a Diagram 4.G.A.1

The two slanted lines meet at the crossing point and continue down to line q, cutting off a triangle whose apex is the crossing point and whose base lies on q. The angle a is the apex angle.

a = 180^\circ - (\text{base angle}_1) - (\text{base angle}_2)

Drawing the picture turns 'angles between parallel lines' into a single triangle whose angles must add to 180 deg.

2Move the 55 deg down to line q

#7 Identify Subproblems 4.G.A.2

A transversal crosses the parallel lines p and q. The 55 deg angle it makes with p equals its alternate interior angle with q, so one base angle of the triangle on q is 55 deg.

55^\circ \text{ at } p = 55^\circ \text{ base angle at } q

Because p and q are parallel, a slanted line crosses them at the same tilt, so the matching (alternate) angle is also the same.

3Use the two base angles

#7 Identify Subproblems 4.MD.C.7

The triangle's two base angles on q are 55 deg and 40 deg. The apex angle a is 180 deg minus their sum.

a = 180^\circ - 55^\circ - 40^\circ = 85^\circ

Once two angles of a triangle are known, the third is forced, because all three always total 180 deg.

Answer: 85 degrees

4 · Reviewdoes it hold up?

The base angles 55 deg and 40 deg add to 95 deg, leaving 85 deg for the apex. All three angles 55 + 40 + 85 = 180 deg.

Another way: Use the exterior angle idea (tool 7): the angle the crossing makes on the p-side is the exterior angle of the same triangle and equals 55 + 40 = 95 deg; the angle a is its straight-line partner, 180 - 95 = 85 deg.

Standardsmin grade 4

4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Identifying the triangle bounded by the two transversals and line q.
4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Using the parallel lines to move the 55 deg angle to its alternate interior angle on q.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting 55 and 40 from 180 to get the apex angle.

💡Takeaway. Find the hidden triangle, slide the top angle down between the parallel lines, and the last angle is just what is left of 180 deg!

Variant 4 medium

In the figure, line $p$ and line $q$ are parallel to each other. Find the measure of angle $a$ .

[Figure] Two horizontal parallel lines, $p$ (top) and $q$ (bottom), are crossed by two transversal lines that intersect in an X shape. Where a transversal meets the upper line $p$ , one angle is marked as $50^\circ$ . Where they meet the lower line $q$ , one angle is marked as $60^\circ$ , and the angle $a$ is marked at the crossing point between the lines.

Your answer

Show solution

1 · Understandwhat's really being asked

Two parallel lines p (top) and q (bottom) are crossed by two slanted lines that meet in an X between them. One angle where a transversal meets the top line p is 50 deg; one angle where a transversal meets the bottom line q is 60 deg. At the crossing point between the lines, the angle a that opens downward toward q is marked. I need that angle a.

Givens

Line p and line q are parallel.
Two transversal lines cross between p and q, forming a triangle whose third side lies along q.
An angle made with line p is 50 deg.
A base angle made with line q is 60 deg.
The angle a is the apex angle of the triangle, at the crossing point, opening toward q.

Unknowns

The measure of the angle a at the crossing point.

Constraints

Alternate interior angles between parallel lines are equal.
The three interior angles of a triangle add to 180 deg.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

The two transversals and line q bound a triangle. Its apex is the crossing point (the angle a), and its base sits on q. The 60 deg is one base angle directly. The 50 deg at line p moves down to the other base angle on q by alternate interior angles (p is parallel to q). Then the apex angle is what is left to reach 180 deg.

3 · Execute3 carry out the plan

1Spot the triangle

#1 Draw a Diagram 4.G.A.1

The two slanted lines meet at the crossing point and continue down to line q, cutting off a triangle whose apex is the crossing point and whose base lies on q. The angle a is the apex angle.

a = 180^\circ - (\text{base angle}_1) - (\text{base angle}_2)

Drawing the picture turns 'angles between parallel lines' into a single triangle whose angles must add to 180 deg.

2Move the 50 deg down to line q

#7 Identify Subproblems 4.G.A.2

A transversal crosses the parallel lines p and q. The 50 deg angle it makes with p equals its alternate interior angle with q, so one base angle of the triangle on q is 50 deg.

50^\circ \text{ at } p = 50^\circ \text{ base angle at } q

Because p and q are parallel, a slanted line crosses them at the same tilt, so the matching (alternate) angle is also the same.

3Use the two base angles

#7 Identify Subproblems 4.MD.C.7

The triangle's two base angles on q are 50 deg and 60 deg. The apex angle a is 180 deg minus their sum.

a = 180^\circ - 50^\circ - 60^\circ = 70^\circ

Once two angles of a triangle are known, the third is forced, because all three always total 180 deg.

Answer: 70 degrees

4 · Reviewdoes it hold up?

The base angles 50 deg and 60 deg add to 110 deg, leaving 70 deg for the apex. All three angles 50 + 60 + 70 = 180 deg.

Another way: Use the exterior angle idea (tool 7): the angle the crossing makes on the p-side is the exterior angle of the same triangle and equals 50 + 60 = 110 deg; the angle a is its straight-line partner, 180 - 110 = 70 deg.

Standardsmin grade 4

4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Identifying the triangle bounded by the two transversals and line q.
4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Using the parallel lines to move the 50 deg angle to its alternate interior angle on q.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting 50 and 60 from 180 to get the apex angle.

💡Takeaway. Find the hidden triangle, slide the top angle down between the parallel lines, and the last angle is just what is left of 180 deg!

Variant 5 medium

In the figure, line $p$ and line $q$ are parallel to each other. Find the measure of angle $a$ .

[Figure] Two horizontal parallel lines, $p$ (top) and $q$ (bottom), are crossed by two transversal lines that intersect in an X shape. Where a transversal meets the upper line $p$ , one angle is marked as $60^\circ$ . Where they meet the lower line $q$ , one angle is marked as $50^\circ$ , and the angle $a$ is marked at the crossing point between the lines.

Your answer

Show solution

1 · Understandwhat's really being asked

Two parallel lines p (top) and q (bottom) are crossed by two slanted lines that meet in an X between them. One angle where a transversal meets the top line p is 60 deg; one angle where a transversal meets the bottom line q is 50 deg. At the crossing point between the lines, the angle a that opens downward toward q is marked. I need that angle a.

Givens

Line p and line q are parallel.
Two transversal lines cross between p and q, forming a triangle whose third side lies along q.
An angle made with line p is 60 deg.
A base angle made with line q is 50 deg.
The angle a is the apex angle of the triangle, at the crossing point, opening toward q.

Unknowns

The measure of the angle a at the crossing point.

Constraints

Alternate interior angles between parallel lines are equal.
The three interior angles of a triangle add to 180 deg.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

The two transversals and line q bound a triangle. Its apex is the crossing point (the angle a), and its base sits on q. The 50 deg is one base angle directly. The 60 deg at line p moves down to the other base angle on q by alternate interior angles (p is parallel to q). Then the apex angle is what is left to reach 180 deg.

3 · Execute3 carry out the plan

1Spot the triangle

#1 Draw a Diagram 4.G.A.1

The two slanted lines meet at the crossing point and continue down to line q, cutting off a triangle whose apex is the crossing point and whose base lies on q. The angle a is the apex angle.

a = 180^\circ - (\text{base angle}_1) - (\text{base angle}_2)

Drawing the picture turns 'angles between parallel lines' into a single triangle whose angles must add to 180 deg.

2Move the 60 deg down to line q

#7 Identify Subproblems 4.G.A.2

A transversal crosses the parallel lines p and q. The 60 deg angle it makes with p equals its alternate interior angle with q, so one base angle of the triangle on q is 60 deg.

60^\circ \text{ at } p = 60^\circ \text{ base angle at } q

Because p and q are parallel, a slanted line crosses them at the same tilt, so the matching (alternate) angle is also the same.

3Use the two base angles

#7 Identify Subproblems 4.MD.C.7

The triangle's two base angles on q are 60 deg and 50 deg. The apex angle a is 180 deg minus their sum.

a = 180^\circ - 60^\circ - 50^\circ = 70^\circ

Once two angles of a triangle are known, the third is forced, because all three always total 180 deg.

Answer: 70 degrees

4 · Reviewdoes it hold up?

The base angles 60 deg and 50 deg add to 110 deg, leaving 70 deg for the apex. All three angles 60 + 50 + 70 = 180 deg.

Another way: Use the exterior angle idea (tool 7): the angle the crossing makes on the p-side is the exterior angle of the same triangle and equals 60 + 50 = 110 deg; the angle a is its straight-line partner, 180 - 110 = 70 deg.

Standardsmin grade 4

4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Identifying the triangle bounded by the two transversals and line q.
4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Using the parallel lines to move the 60 deg angle to its alternate interior angle on q.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting 60 and 50 from 180 to get the apex angle.

💡Takeaway. Find the hidden triangle, slide the top angle down between the parallel lines, and the last angle is just what is left of 180 deg!

Variant 6 medium

In the figure, line $p$ and line $q$ are parallel to each other. Find the measure of angle $a$ .

[Figure] Two horizontal parallel lines, $p$ (top) and $q$ (bottom), are crossed by two transversal lines that intersect in an X shape. Where a transversal meets the upper line $p$ , one angle is marked as $65^\circ$ . Where they meet the lower line $q$ , one angle is marked as $45^\circ$ , and the angle $a$ is marked at the crossing point between the lines.

Your answer

Show solution

1 · Understandwhat's really being asked

Two parallel lines p (top) and q (bottom) are crossed by two slanted lines that meet in an X between them. One angle where a transversal meets the top line p is 65 deg; one angle where a transversal meets the bottom line q is 45 deg. At the crossing point between the lines, the angle a that opens downward toward q is marked. I need that angle a.

Givens

Line p and line q are parallel.
Two transversal lines cross between p and q, forming a triangle whose third side lies along q.
An angle made with line p is 65 deg.
A base angle made with line q is 45 deg.
The angle a is the apex angle of the triangle, at the crossing point, opening toward q.

Unknowns

The measure of the angle a at the crossing point.

Constraints

Alternate interior angles between parallel lines are equal.
The three interior angles of a triangle add to 180 deg.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

The two transversals and line q bound a triangle. Its apex is the crossing point (the angle a), and its base sits on q. The 45 deg is one base angle directly. The 65 deg at line p moves down to the other base angle on q by alternate interior angles (p is parallel to q). Then the apex angle is what is left to reach 180 deg.

3 · Execute3 carry out the plan

1Spot the triangle

#1 Draw a Diagram 4.G.A.1

The two slanted lines meet at the crossing point and continue down to line q, cutting off a triangle whose apex is the crossing point and whose base lies on q. The angle a is the apex angle.

a = 180^\circ - (\text{base angle}_1) - (\text{base angle}_2)

Drawing the picture turns 'angles between parallel lines' into a single triangle whose angles must add to 180 deg.

2Move the 65 deg down to line q

#7 Identify Subproblems 4.G.A.2

A transversal crosses the parallel lines p and q. The 65 deg angle it makes with p equals its alternate interior angle with q, so one base angle of the triangle on q is 65 deg.

65^\circ \text{ at } p = 65^\circ \text{ base angle at } q

Because p and q are parallel, a slanted line crosses them at the same tilt, so the matching (alternate) angle is also the same.

3Use the two base angles

#7 Identify Subproblems 4.MD.C.7

The triangle's two base angles on q are 65 deg and 45 deg. The apex angle a is 180 deg minus their sum.

a = 180^\circ - 65^\circ - 45^\circ = 70^\circ

Once two angles of a triangle are known, the third is forced, because all three always total 180 deg.

Answer: 70 degrees

4 · Reviewdoes it hold up?

The base angles 65 deg and 45 deg add to 110 deg, leaving 70 deg for the apex. All three angles 65 + 45 + 70 = 180 deg.

Another way: Use the exterior angle idea (tool 7): the angle the crossing makes on the p-side is the exterior angle of the same triangle and equals 65 + 45 = 110 deg; the angle a is its straight-line partner, 180 - 110 = 70 deg.

Standardsmin grade 4

4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Identifying the triangle bounded by the two transversals and line q.
4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Using the parallel lines to move the 65 deg angle to its alternate interior angle on q.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting 65 and 45 from 180 to get the apex angle.

💡Takeaway. Find the hidden triangle, slide the top angle down between the parallel lines, and the last angle is just what is left of 180 deg!

Variant 7 medium

In the figure, line $p$ and line $q$ are parallel to each other. Find the measure of angle $a$ .

[Figure] Two horizontal parallel lines, $p$ (top) and $q$ (bottom), are crossed by two transversal lines that intersect in an X shape. Where a transversal meets the upper line $p$ , one angle is marked as $35^\circ$ . Where they meet the lower line $q$ , one angle is marked as $65^\circ$ , and the angle $a$ is marked at the crossing point between the lines.

Your answer

Show solution

1 · Understandwhat's really being asked

Two parallel lines p (top) and q (bottom) are crossed by two slanted lines that meet in an X between them. One angle where a transversal meets the top line p is 35 deg; one angle where a transversal meets the bottom line q is 65 deg. At the crossing point between the lines, the angle a that opens downward toward q is marked. I need that angle a.

Givens

Line p and line q are parallel.
Two transversal lines cross between p and q, forming a triangle whose third side lies along q.
An angle made with line p is 35 deg.
A base angle made with line q is 65 deg.
The angle a is the apex angle of the triangle, at the crossing point, opening toward q.

Unknowns

The measure of the angle a at the crossing point.

Constraints

Alternate interior angles between parallel lines are equal.
The three interior angles of a triangle add to 180 deg.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

The two transversals and line q bound a triangle. Its apex is the crossing point (the angle a), and its base sits on q. The 65 deg is one base angle directly. The 35 deg at line p moves down to the other base angle on q by alternate interior angles (p is parallel to q). Then the apex angle is what is left to reach 180 deg.

3 · Execute3 carry out the plan

1Spot the triangle

#1 Draw a Diagram 4.G.A.1

The two slanted lines meet at the crossing point and continue down to line q, cutting off a triangle whose apex is the crossing point and whose base lies on q. The angle a is the apex angle.

a = 180^\circ - (\text{base angle}_1) - (\text{base angle}_2)

Drawing the picture turns 'angles between parallel lines' into a single triangle whose angles must add to 180 deg.

2Move the 35 deg down to line q

#7 Identify Subproblems 4.G.A.2

A transversal crosses the parallel lines p and q. The 35 deg angle it makes with p equals its alternate interior angle with q, so one base angle of the triangle on q is 35 deg.

35^\circ \text{ at } p = 35^\circ \text{ base angle at } q

Because p and q are parallel, a slanted line crosses them at the same tilt, so the matching (alternate) angle is also the same.

3Use the two base angles

#7 Identify Subproblems 4.MD.C.7

The triangle's two base angles on q are 35 deg and 65 deg. The apex angle a is 180 deg minus their sum.

a = 180^\circ - 35^\circ - 65^\circ = 80^\circ

Once two angles of a triangle are known, the third is forced, because all three always total 180 deg.

Answer: 80 degrees

4 · Reviewdoes it hold up?

The base angles 35 deg and 65 deg add to 100 deg, leaving 80 deg for the apex. All three angles 35 + 65 + 80 = 180 deg.

Another way: Use the exterior angle idea (tool 7): the angle the crossing makes on the p-side is the exterior angle of the same triangle and equals 35 + 65 = 100 deg; the angle a is its straight-line partner, 180 - 100 = 80 deg.

Standardsmin grade 4

4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Identifying the triangle bounded by the two transversals and line q.
4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Using the parallel lines to move the 35 deg angle to its alternate interior angle on q.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting 35 and 65 from 180 to get the apex angle.

💡Takeaway. Find the hidden triangle, slide the top angle down between the parallel lines, and the last angle is just what is left of 180 deg!

Variant 8 hard

In the figure, line $p$ and line $q$ are parallel to each other. Find the measure of angle $a$ .

[Figure] Two horizontal parallel lines, $p$ (top) and $q$ (bottom), are crossed by two transversal lines that intersect in an X shape. Where a transversal meets the upper line $p$ , one angle is marked as $70^\circ$ . Where they meet the lower line $q$ , one angle is marked as $30^\circ$ , and the angle $a$ is marked at the crossing point between the lines.

Your answer

Show solution

1 · Understandwhat's really being asked

Two parallel lines p (top) and q (bottom) are crossed by two slanted lines that meet in an X between them. One angle where a transversal meets the top line p is 70 deg; one angle where a transversal meets the bottom line q is 30 deg. At the crossing point between the lines, the angle a that opens downward toward q is marked. I need that angle a.

Givens

Line p and line q are parallel.
Two transversal lines cross between p and q, forming a triangle whose third side lies along q.
An angle made with line p is 70 deg.
A base angle made with line q is 30 deg.
The angle a is the apex angle of the triangle, at the crossing point, opening toward q.

Unknowns

The measure of the angle a at the crossing point.

Constraints

Alternate interior angles between parallel lines are equal.
The three interior angles of a triangle add to 180 deg.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

The two transversals and line q bound a triangle. Its apex is the crossing point (the angle a), and its base sits on q. The 30 deg is one base angle directly. The 70 deg at line p moves down to the other base angle on q by alternate interior angles (p is parallel to q). Then the apex angle is what is left to reach 180 deg.

3 · Execute3 carry out the plan

1Spot the triangle

#1 Draw a Diagram 4.G.A.1

The two slanted lines meet at the crossing point and continue down to line q, cutting off a triangle whose apex is the crossing point and whose base lies on q. The angle a is the apex angle.

a = 180^\circ - (\text{base angle}_1) - (\text{base angle}_2)

Drawing the picture turns 'angles between parallel lines' into a single triangle whose angles must add to 180 deg.

2Move the 70 deg down to line q

#7 Identify Subproblems 4.G.A.2

A transversal crosses the parallel lines p and q. The 70 deg angle it makes with p equals its alternate interior angle with q, so one base angle of the triangle on q is 70 deg.

70^\circ \text{ at } p = 70^\circ \text{ base angle at } q

Because p and q are parallel, a slanted line crosses them at the same tilt, so the matching (alternate) angle is also the same.

3Use the two base angles

#7 Identify Subproblems 4.MD.C.7

The triangle's two base angles on q are 70 deg and 30 deg. The apex angle a is 180 deg minus their sum.

a = 180^\circ - 70^\circ - 30^\circ = 80^\circ

Once two angles of a triangle are known, the third is forced, because all three always total 180 deg.

Answer: 80 degrees

4 · Reviewdoes it hold up?

The base angles 70 deg and 30 deg add to 100 deg, leaving 80 deg for the apex. All three angles 70 + 30 + 80 = 180 deg.

Another way: Use the exterior angle idea (tool 7): the angle the crossing makes on the p-side is the exterior angle of the same triangle and equals 70 + 30 = 100 deg; the angle a is its straight-line partner, 180 - 100 = 80 deg.

Standardsmin grade 4

4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Identifying the triangle bounded by the two transversals and line q.
4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Using the parallel lines to move the 70 deg angle to its alternate interior angle on q.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting 70 and 30 from 180 to get the apex angle.

💡Takeaway. Find the hidden triangle, slide the top angle down between the parallel lines, and the last angle is just what is left of 180 deg!

Variant 9 hard

In the figure, line $p$ and line $q$ are parallel to each other. Find the measure of angle $a$ .

[Figure] Two horizontal parallel lines, $p$ (top) and $q$ (bottom), are crossed by two transversal lines that intersect in an X shape. Where a transversal meets the upper line $p$ , one angle is marked as $25^\circ$ . Where they meet the lower line $q$ , one angle is marked as $75^\circ$ , and the angle $a$ is marked at the crossing point between the lines.

Your answer

Show solution

1 · Understandwhat's really being asked

Two parallel lines p (top) and q (bottom) are crossed by two slanted lines that meet in an X between them. One angle where a transversal meets the top line p is 25 deg; one angle where a transversal meets the bottom line q is 75 deg. At the crossing point between the lines, the angle a that opens downward toward q is marked. I need that angle a.

Givens

Line p and line q are parallel.
Two transversal lines cross between p and q, forming a triangle whose third side lies along q.
An angle made with line p is 25 deg.
A base angle made with line q is 75 deg.
The angle a is the apex angle of the triangle, at the crossing point, opening toward q.

Unknowns

The measure of the angle a at the crossing point.

Constraints

Alternate interior angles between parallel lines are equal.
The three interior angles of a triangle add to 180 deg.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

The two transversals and line q bound a triangle. Its apex is the crossing point (the angle a), and its base sits on q. The 75 deg is one base angle directly. The 25 deg at line p moves down to the other base angle on q by alternate interior angles (p is parallel to q). Then the apex angle is what is left to reach 180 deg.

3 · Execute3 carry out the plan

1Spot the triangle

#1 Draw a Diagram 4.G.A.1

The two slanted lines meet at the crossing point and continue down to line q, cutting off a triangle whose apex is the crossing point and whose base lies on q. The angle a is the apex angle.

a = 180^\circ - (\text{base angle}_1) - (\text{base angle}_2)

Drawing the picture turns 'angles between parallel lines' into a single triangle whose angles must add to 180 deg.

2Move the 25 deg down to line q

#7 Identify Subproblems 4.G.A.2

A transversal crosses the parallel lines p and q. The 25 deg angle it makes with p equals its alternate interior angle with q, so one base angle of the triangle on q is 25 deg.

25^\circ \text{ at } p = 25^\circ \text{ base angle at } q

Because p and q are parallel, a slanted line crosses them at the same tilt, so the matching (alternate) angle is also the same.

3Use the two base angles

#7 Identify Subproblems 4.MD.C.7

The triangle's two base angles on q are 25 deg and 75 deg. The apex angle a is 180 deg minus their sum.

a = 180^\circ - 25^\circ - 75^\circ = 80^\circ

Once two angles of a triangle are known, the third is forced, because all three always total 180 deg.

Answer: 80 degrees

4 · Reviewdoes it hold up?

The base angles 25 deg and 75 deg add to 100 deg, leaving 80 deg for the apex. All three angles 25 + 75 + 80 = 180 deg.

Another way: Use the exterior angle idea (tool 7): the angle the crossing makes on the p-side is the exterior angle of the same triangle and equals 25 + 75 = 100 deg; the angle a is its straight-line partner, 180 - 100 = 80 deg.

Standardsmin grade 4

4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Identifying the triangle bounded by the two transversals and line q.
4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Using the parallel lines to move the 25 deg angle to its alternate interior angle on q.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting 25 and 75 from 180 to get the apex angle.

💡Takeaway. Find the hidden triangle, slide the top angle down between the parallel lines, and the last angle is just what is left of 180 deg!

Variant 10 hard

In the figure, line $p$ and line $q$ are parallel to each other. Find the measure of angle $a$ .

[Figure] Two horizontal parallel lines, $p$ (top) and $q$ (bottom), are crossed by two transversal lines that intersect in an X shape. Where a transversal meets the upper line $p$ , one angle is marked as $80^\circ$ . Where they meet the lower line $q$ , one angle is marked as $25^\circ$ , and the angle $a$ is marked at the crossing point between the lines.

Your answer

Show solution

1 · Understandwhat's really being asked

Two parallel lines p (top) and q (bottom) are crossed by two slanted lines that meet in an X between them. One angle where a transversal meets the top line p is 80 deg; one angle where a transversal meets the bottom line q is 25 deg. At the crossing point between the lines, the angle a that opens downward toward q is marked. I need that angle a.

Givens

Line p and line q are parallel.
Two transversal lines cross between p and q, forming a triangle whose third side lies along q.
An angle made with line p is 80 deg.
A base angle made with line q is 25 deg.
The angle a is the apex angle of the triangle, at the crossing point, opening toward q.

Unknowns

The measure of the angle a at the crossing point.

Constraints

Alternate interior angles between parallel lines are equal.
The three interior angles of a triangle add to 180 deg.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

The two transversals and line q bound a triangle. Its apex is the crossing point (the angle a), and its base sits on q. The 25 deg is one base angle directly. The 80 deg at line p moves down to the other base angle on q by alternate interior angles (p is parallel to q). Then the apex angle is what is left to reach 180 deg.

3 · Execute3 carry out the plan

1Spot the triangle

#1 Draw a Diagram 4.G.A.1

The two slanted lines meet at the crossing point and continue down to line q, cutting off a triangle whose apex is the crossing point and whose base lies on q. The angle a is the apex angle.

a = 180^\circ - (\text{base angle}_1) - (\text{base angle}_2)

Drawing the picture turns 'angles between parallel lines' into a single triangle whose angles must add to 180 deg.

2Move the 80 deg down to line q

#7 Identify Subproblems 4.G.A.2

A transversal crosses the parallel lines p and q. The 80 deg angle it makes with p equals its alternate interior angle with q, so one base angle of the triangle on q is 80 deg.

80^\circ \text{ at } p = 80^\circ \text{ base angle at } q

Because p and q are parallel, a slanted line crosses them at the same tilt, so the matching (alternate) angle is also the same.

3Use the two base angles

#7 Identify Subproblems 4.MD.C.7

The triangle's two base angles on q are 80 deg and 25 deg. The apex angle a is 180 deg minus their sum.

a = 180^\circ - 80^\circ - 25^\circ = 75^\circ

Once two angles of a triangle are known, the third is forced, because all three always total 180 deg.

Answer: 75 degrees

4 · Reviewdoes it hold up?

The base angles 80 deg and 25 deg add to 105 deg, leaving 75 deg for the apex. All three angles 80 + 25 + 75 = 180 deg.

Another way: Use the exterior angle idea (tool 7): the angle the crossing makes on the p-side is the exterior angle of the same triangle and equals 80 + 25 = 105 deg; the angle a is its straight-line partner, 180 - 105 = 75 deg.

Standardsmin grade 4

4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Identifying the triangle bounded by the two transversals and line q.
4.G.A.2 Classify two-dimensional figures based on presence of parallel or perpendicular lines — Using the parallel lines to move the 80 deg angle to its alternate interior angle on q.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting 80 and 25 from 180 to get the apex angle.

💡Takeaway. Find the hidden triangle, slide the top angle down between the parallel lines, and the last angle is just what is left of 180 deg!

Where next?

← Back to the problem: Corresponding and alternate angles are equal