← 4-1 · Triangle angles sum to 180 degrees · Angle Facts in a Figure

Triangle angles sum to 180 degrees · 10 practice problems

4.MD.C.7

Generated variants — 10

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

Variant 1 answer: 50 degrees

Using the fact that the three angles of a triangle add up to $180^\circ$ , find the measure of angle $a$ in the figure.

[Figure] Triangle $ABD$ has a segment $AC$ drawn from vertex $A$ down to point $C$ on the base $BD$ , splitting it into two smaller triangles. At point $C$ , the angle on the left-triangle side ( $\angle ACB$ ) is $110^\circ$ , and at vertex $A$ the angle of the right-hand triangle ( $\angle CAD$ ) is $60^\circ$ . The angle $a$ to be found is at vertex $D$ .

Show solution

Understand

Big triangle ABD has a line AC drawn from the top vertex A down to point C on the base BD, making two smaller triangles. At C, the left-triangle angle ACB is 110 degrees. At A, the right-triangle angle CAD is 60 degrees. Find angle a at vertex D.

Givens

Triangle ABD with point C on base BD and segment AC drawn.
Angle ACB = 110 degrees (left triangle, at C).
Angle CAD = 60 degrees (right triangle, at A).
Angle a is at vertex D in the right triangle ACD.

Unknowns

The measure of angle a at vertex D.

Constraints

The three angles of any triangle add to 180 degrees.
Angles ACB and ACD sit on the straight line BD, so they add to 180 degrees.

Plan

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

Subproblem 1: angle ACD is the straight-line partner of the 110-degree angle, so 180 - 110. Subproblem 2: in the right triangle ACD the three angles add to 180 degrees, so a = 180 - (CAD) - (ACD).

Execute

#7 Identify Subproblems 4.MD.C.7

Point C is on the straight line BD, so the angle on the left (ACB = 110 degrees) and the angle on the right (ACD) together make the straight angle 180 degrees.

\angle ACD = 180^\circ - 110^\circ = 70^\circ

The base is a flat 180-degree line split into the two angles at C.

#7 Identify Subproblems 4.MD.C.7

In the right-hand triangle ACD the three angles are CAD = 60 degrees, ACD = 70 degrees, and a at D. They add to 180 degrees, so subtract the two known angles.

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

Every triangle's three corners always total 180 degrees, so the third corner is the leftover.

Answer: 50 degrees

Review

Triangle ACD: 60 + 70 + 50 = 180 degrees, a valid triangle. The straight base at C: 110 + 70 = 180 degrees. Both checks hold, so a = 50 degrees.

Use the exterior-angle idea (tool 5/pattern): the 110-degree angle ACB is the outside angle of triangle ACD at C, and it equals the two far angles CAD + a, so 110 = 60 + a gives a = 50 degrees.

Standards · min grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Using the 180-degree straight line and the 180-degree triangle total to find a.
4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Reading the triangle, the base point C, and segment AC in the figure.

💡 Split the base with the 180-degree line, then the triangle's 180-degree rule hands you angle a - pure Grade 4 angle adding!

Variant 2 answer: 70 degrees

Using the fact that the three angles of a triangle add up to $180^\circ$ , find the measure of angle $a$ in the figure.

[Figure] Triangle $ABD$ has a segment $AC$ drawn from vertex $A$ down to point $C$ on the base $BD$ , splitting it into two smaller triangles. At point $C$ , the angle on the left-triangle side ( $\angle ACB$ ) is $105^\circ$ , and at vertex $A$ the angle of the right-hand triangle ( $\angle CAD$ ) is $35^\circ$ . The angle $a$ to be found is at vertex $D$ .

Show solution

Understand

Big triangle ABD has a line AC drawn from the top vertex A down to point C on the base BD, making two smaller triangles. At C, the left-triangle angle ACB is 105 degrees. At A, the right-triangle angle CAD is 35 degrees. Find angle a at vertex D.

Givens

Triangle ABD with point C on base BD and segment AC drawn.
Angle ACB = 105 degrees (left triangle, at C).
Angle CAD = 35 degrees (right triangle, at A).
Angle a is at vertex D in the right triangle ACD.

Unknowns

The measure of angle a at vertex D.

Constraints

The three angles of any triangle add to 180 degrees.
Angles ACB and ACD sit on the straight line BD, so they add to 180 degrees.

Plan

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

Subproblem 1: angle ACD is the straight-line partner of the 105-degree angle, so 180 - 105. Subproblem 2: in the right triangle ACD the three angles add to 180 degrees, so a = 180 - (CAD) - (ACD).

Execute

#7 Identify Subproblems 4.MD.C.7

Point C is on the straight line BD, so the angle on the left (ACB = 105 degrees) and the angle on the right (ACD) together make the straight angle 180 degrees.

\angle ACD = 180^\circ - 105^\circ = 75^\circ

The base is a flat 180-degree line split into the two angles at C.

#7 Identify Subproblems 4.MD.C.7

In the right-hand triangle ACD the three angles are CAD = 35 degrees, ACD = 75 degrees, and a at D. They add to 180 degrees, so subtract the two known angles.

a = 180^\circ - 35^\circ - 75^\circ = 70^\circ

Every triangle's three corners always total 180 degrees, so the third corner is the leftover.

Answer: 70 degrees

Review

Triangle ACD: 35 + 75 + 70 = 180 degrees, a valid triangle. The straight base at C: 105 + 75 = 180 degrees. Both checks hold, so a = 70 degrees.

Use the exterior-angle idea (tool 5/pattern): the 105-degree angle ACB is the outside angle of triangle ACD at C, and it equals the two far angles CAD + a, so 105 = 35 + a gives a = 70 degrees.

Standards · min grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Using the 180-degree straight line and the 180-degree triangle total to find a.
4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Reading the triangle, the base point C, and segment AC in the figure.

💡 Split the base with the 180-degree line, then the triangle's 180-degree rule hands you angle a - pure Grade 4 angle adding!

Variant 3 answer: 60 degrees

Using the fact that the three angles of a triangle add up to $180^\circ$ , find the measure of angle $a$ in the figure.

[Figure] Triangle $ABD$ has a segment $AC$ drawn from vertex $A$ down to point $C$ on the base $BD$ , splitting it into two smaller triangles. At point $C$ , the angle on the left-triangle side ( $\angle ACB$ ) is $140^\circ$ , and at vertex $A$ the angle of the right-hand triangle ( $\angle CAD$ ) is $80^\circ$ . The angle $a$ to be found is at vertex $D$ .

Show solution

Understand

Big triangle ABD has a line AC drawn from the top vertex A down to point C on the base BD, making two smaller triangles. At C, the left-triangle angle ACB is 140 degrees. At A, the right-triangle angle CAD is 80 degrees. Find angle a at vertex D.

Givens

Triangle ABD with point C on base BD and segment AC drawn.
Angle ACB = 140 degrees (left triangle, at C).
Angle CAD = 80 degrees (right triangle, at A).
Angle a is at vertex D in the right triangle ACD.

Unknowns

The measure of angle a at vertex D.

Constraints

The three angles of any triangle add to 180 degrees.
Angles ACB and ACD sit on the straight line BD, so they add to 180 degrees.

Plan

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

Subproblem 1: angle ACD is the straight-line partner of the 140-degree angle, so 180 - 140. Subproblem 2: in the right triangle ACD the three angles add to 180 degrees, so a = 180 - (CAD) - (ACD).

Execute

#7 Identify Subproblems 4.MD.C.7

Point C is on the straight line BD, so the angle on the left (ACB = 140 degrees) and the angle on the right (ACD) together make the straight angle 180 degrees.

\angle ACD = 180^\circ - 140^\circ = 40^\circ

The base is a flat 180-degree line split into the two angles at C.

#7 Identify Subproblems 4.MD.C.7

In the right-hand triangle ACD the three angles are CAD = 80 degrees, ACD = 40 degrees, and a at D. They add to 180 degrees, so subtract the two known angles.

a = 180^\circ - 80^\circ - 40^\circ = 60^\circ

Every triangle's three corners always total 180 degrees, so the third corner is the leftover.

Answer: 60 degrees

Review

Triangle ACD: 80 + 40 + 60 = 180 degrees, a valid triangle. The straight base at C: 140 + 40 = 180 degrees. Both checks hold, so a = 60 degrees.

Use the exterior-angle idea (tool 5/pattern): the 140-degree angle ACB is the outside angle of triangle ACD at C, and it equals the two far angles CAD + a, so 140 = 80 + a gives a = 60 degrees.

Standards · min grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Using the 180-degree straight line and the 180-degree triangle total to find a.
4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Reading the triangle, the base point C, and segment AC in the figure.

💡 Split the base with the 180-degree line, then the triangle's 180-degree rule hands you angle a - pure Grade 4 angle adding!

Variant 4 answer: 50 degrees

Using the fact that the three angles of a triangle add up to $180^\circ$ , find the measure of angle $a$ in the figure.

[Figure] Triangle $ABD$ has a segment $AC$ drawn from vertex $A$ down to point $C$ on the base $BD$ , splitting it into two smaller triangles. At point $C$ , the angle on the left-triangle side ( $\angle ACB$ ) is $120^\circ$ , and at vertex $A$ the angle of the right-hand triangle ( $\angle CAD$ ) is $70^\circ$ . The angle $a$ to be found is at vertex $D$ .

Show solution

Understand

Big triangle ABD has a line AC drawn from the top vertex A down to point C on the base BD, making two smaller triangles. At C, the left-triangle angle ACB is 120 degrees. At A, the right-triangle angle CAD is 70 degrees. Find angle a at vertex D.

Givens

Triangle ABD with point C on base BD and segment AC drawn.
Angle ACB = 120 degrees (left triangle, at C).
Angle CAD = 70 degrees (right triangle, at A).
Angle a is at vertex D in the right triangle ACD.

Unknowns

The measure of angle a at vertex D.

Constraints

The three angles of any triangle add to 180 degrees.
Angles ACB and ACD sit on the straight line BD, so they add to 180 degrees.

Plan

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

Subproblem 1: angle ACD is the straight-line partner of the 120-degree angle, so 180 - 120. Subproblem 2: in the right triangle ACD the three angles add to 180 degrees, so a = 180 - (CAD) - (ACD).

Execute

#7 Identify Subproblems 4.MD.C.7

Point C is on the straight line BD, so the angle on the left (ACB = 120 degrees) and the angle on the right (ACD) together make the straight angle 180 degrees.

\angle ACD = 180^\circ - 120^\circ = 60^\circ

The base is a flat 180-degree line split into the two angles at C.

#7 Identify Subproblems 4.MD.C.7

In the right-hand triangle ACD the three angles are CAD = 70 degrees, ACD = 60 degrees, and a at D. They add to 180 degrees, so subtract the two known angles.

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

Every triangle's three corners always total 180 degrees, so the third corner is the leftover.

Answer: 50 degrees

Review

Triangle ACD: 70 + 60 + 50 = 180 degrees, a valid triangle. The straight base at C: 120 + 60 = 180 degrees. Both checks hold, so a = 50 degrees.

Use the exterior-angle idea (tool 5/pattern): the 120-degree angle ACB is the outside angle of triangle ACD at C, and it equals the two far angles CAD + a, so 120 = 70 + a gives a = 50 degrees.

Standards · min grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Using the 180-degree straight line and the 180-degree triangle total to find a.
4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Reading the triangle, the base point C, and segment AC in the figure.

💡 Split the base with the 180-degree line, then the triangle's 180-degree rule hands you angle a - pure Grade 4 angle adding!

Variant 5 answer: 50 degrees

Using the fact that the three angles of a triangle add up to $180^\circ$ , find the measure of angle $a$ in the figure.

[Figure] Triangle $ABD$ has a segment $AC$ drawn from vertex $A$ down to point $C$ on the base $BD$ , splitting it into two smaller triangles. At point $C$ , the angle on the left-triangle side ( $\angle ACB$ ) is $115^\circ$ , and at vertex $A$ the angle of the right-hand triangle ( $\angle CAD$ ) is $65^\circ$ . The angle $a$ to be found is at vertex $D$ .

Show solution

Understand

Big triangle ABD has a line AC drawn from the top vertex A down to point C on the base BD, making two smaller triangles. At C, the left-triangle angle ACB is 115 degrees. At A, the right-triangle angle CAD is 65 degrees. Find angle a at vertex D.

Givens

Triangle ABD with point C on base BD and segment AC drawn.
Angle ACB = 115 degrees (left triangle, at C).
Angle CAD = 65 degrees (right triangle, at A).
Angle a is at vertex D in the right triangle ACD.

Unknowns

The measure of angle a at vertex D.

Constraints

The three angles of any triangle add to 180 degrees.
Angles ACB and ACD sit on the straight line BD, so they add to 180 degrees.

Plan

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

Subproblem 1: angle ACD is the straight-line partner of the 115-degree angle, so 180 - 115. Subproblem 2: in the right triangle ACD the three angles add to 180 degrees, so a = 180 - (CAD) - (ACD).

Execute

#7 Identify Subproblems 4.MD.C.7

Point C is on the straight line BD, so the angle on the left (ACB = 115 degrees) and the angle on the right (ACD) together make the straight angle 180 degrees.

\angle ACD = 180^\circ - 115^\circ = 65^\circ

The base is a flat 180-degree line split into the two angles at C.

#7 Identify Subproblems 4.MD.C.7

In the right-hand triangle ACD the three angles are CAD = 65 degrees, ACD = 65 degrees, and a at D. They add to 180 degrees, so subtract the two known angles.

a = 180^\circ - 65^\circ - 65^\circ = 50^\circ

Every triangle's three corners always total 180 degrees, so the third corner is the leftover.

Answer: 50 degrees

Review

Triangle ACD: 65 + 65 + 50 = 180 degrees, a valid triangle. The straight base at C: 115 + 65 = 180 degrees. Both checks hold, so a = 50 degrees.

Use the exterior-angle idea (tool 5/pattern): the 115-degree angle ACB is the outside angle of triangle ACD at C, and it equals the two far angles CAD + a, so 115 = 65 + a gives a = 50 degrees.

Standards · min grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Using the 180-degree straight line and the 180-degree triangle total to find a.
4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Reading the triangle, the base point C, and segment AC in the figure.

💡 Split the base with the 180-degree line, then the triangle's 180-degree rule hands you angle a - pure Grade 4 angle adding!

Variant 6 answer: 60 degrees

Using the fact that the three angles of a triangle add up to $180^\circ$ , find the measure of angle $a$ in the figure.

[Figure] Triangle $ABD$ has a segment $AC$ drawn from vertex $A$ down to point $C$ on the base $BD$ , splitting it into two smaller triangles. At point $C$ , the angle on the left-triangle side ( $\angle ACB$ ) is $150^\circ$ , and at vertex $A$ the angle of the right-hand triangle ( $\angle CAD$ ) is $90^\circ$ . The angle $a$ to be found is at vertex $D$ .

Show solution

Understand

Big triangle ABD has a line AC drawn from the top vertex A down to point C on the base BD, making two smaller triangles. At C, the left-triangle angle ACB is 150 degrees. At A, the right-triangle angle CAD is 90 degrees. Find angle a at vertex D.

Givens

Triangle ABD with point C on base BD and segment AC drawn.
Angle ACB = 150 degrees (left triangle, at C).
Angle CAD = 90 degrees (right triangle, at A).
Angle a is at vertex D in the right triangle ACD.

Unknowns

The measure of angle a at vertex D.

Constraints

The three angles of any triangle add to 180 degrees.
Angles ACB and ACD sit on the straight line BD, so they add to 180 degrees.

Plan

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

Subproblem 1: angle ACD is the straight-line partner of the 150-degree angle, so 180 - 150. Subproblem 2: in the right triangle ACD the three angles add to 180 degrees, so a = 180 - (CAD) - (ACD).

Execute

#7 Identify Subproblems 4.MD.C.7

Point C is on the straight line BD, so the angle on the left (ACB = 150 degrees) and the angle on the right (ACD) together make the straight angle 180 degrees.

\angle ACD = 180^\circ - 150^\circ = 30^\circ

The base is a flat 180-degree line split into the two angles at C.

#7 Identify Subproblems 4.MD.C.7

In the right-hand triangle ACD the three angles are CAD = 90 degrees, ACD = 30 degrees, and a at D. They add to 180 degrees, so subtract the two known angles.

a = 180^\circ - 90^\circ - 30^\circ = 60^\circ

Every triangle's three corners always total 180 degrees, so the third corner is the leftover.

Answer: 60 degrees

Review

Triangle ACD: 90 + 30 + 60 = 180 degrees, a valid triangle. The straight base at C: 150 + 30 = 180 degrees. Both checks hold, so a = 60 degrees.

Use the exterior-angle idea (tool 5/pattern): the 150-degree angle ACB is the outside angle of triangle ACD at C, and it equals the two far angles CAD + a, so 150 = 90 + a gives a = 60 degrees.

Standards · min grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Using the 180-degree straight line and the 180-degree triangle total to find a.
4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Reading the triangle, the base point C, and segment AC in the figure.

💡 Split the base with the 180-degree line, then the triangle's 180-degree rule hands you angle a - pure Grade 4 angle adding!

Variant 7 answer: 80 degrees

Using the fact that the three angles of a triangle add up to $180^\circ$ , find the measure of angle $a$ in the figure.

[Figure] Triangle $ABD$ has a segment $AC$ drawn from vertex $A$ down to point $C$ on the base $BD$ , splitting it into two smaller triangles. At point $C$ , the angle on the left-triangle side ( $\angle ACB$ ) is $125^\circ$ , and at vertex $A$ the angle of the right-hand triangle ( $\angle CAD$ ) is $45^\circ$ . The angle $a$ to be found is at vertex $D$ .

Show solution

Understand

Big triangle ABD has a line AC drawn from the top vertex A down to point C on the base BD, making two smaller triangles. At C, the left-triangle angle ACB is 125 degrees. At A, the right-triangle angle CAD is 45 degrees. Find angle a at vertex D.

Givens

Triangle ABD with point C on base BD and segment AC drawn.
Angle ACB = 125 degrees (left triangle, at C).
Angle CAD = 45 degrees (right triangle, at A).
Angle a is at vertex D in the right triangle ACD.

Unknowns

The measure of angle a at vertex D.

Constraints

The three angles of any triangle add to 180 degrees.
Angles ACB and ACD sit on the straight line BD, so they add to 180 degrees.

Plan

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

Subproblem 1: angle ACD is the straight-line partner of the 125-degree angle, so 180 - 125. Subproblem 2: in the right triangle ACD the three angles add to 180 degrees, so a = 180 - (CAD) - (ACD).

Execute

#7 Identify Subproblems 4.MD.C.7

Point C is on the straight line BD, so the angle on the left (ACB = 125 degrees) and the angle on the right (ACD) together make the straight angle 180 degrees.

\angle ACD = 180^\circ - 125^\circ = 55^\circ

The base is a flat 180-degree line split into the two angles at C.

#7 Identify Subproblems 4.MD.C.7

In the right-hand triangle ACD the three angles are CAD = 45 degrees, ACD = 55 degrees, and a at D. They add to 180 degrees, so subtract the two known angles.

a = 180^\circ - 45^\circ - 55^\circ = 80^\circ

Every triangle's three corners always total 180 degrees, so the third corner is the leftover.

Answer: 80 degrees

Review

Triangle ACD: 45 + 55 + 80 = 180 degrees, a valid triangle. The straight base at C: 125 + 55 = 180 degrees. Both checks hold, so a = 80 degrees.

Use the exterior-angle idea (tool 5/pattern): the 125-degree angle ACB is the outside angle of triangle ACD at C, and it equals the two far angles CAD + a, so 125 = 45 + a gives a = 80 degrees.

Standards · min grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Using the 180-degree straight line and the 180-degree triangle total to find a.
4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Reading the triangle, the base point C, and segment AC in the figure.

💡 Split the base with the 180-degree line, then the triangle's 180-degree rule hands you angle a - pure Grade 4 angle adding!

Variant 8 answer: 80 degrees

Using the fact that the three angles of a triangle add up to $180^\circ$ , find the measure of angle $a$ in the figure.

[Figure] Triangle $ABD$ has a segment $AC$ drawn from vertex $A$ down to point $C$ on the base $BD$ , splitting it into two smaller triangles. At point $C$ , the angle on the left-triangle side ( $\angle ACB$ ) is $130^\circ$ , and at vertex $A$ the angle of the right-hand triangle ( $\angle CAD$ ) is $50^\circ$ . The angle $a$ to be found is at vertex $D$ .

Show solution

Understand

Big triangle ABD has a line AC drawn from the top vertex A down to point C on the base BD, making two smaller triangles. At C, the left-triangle angle ACB is 130 degrees. At A, the right-triangle angle CAD is 50 degrees. Find angle a at vertex D.

Givens

Triangle ABD with point C on base BD and segment AC drawn.
Angle ACB = 130 degrees (left triangle, at C).
Angle CAD = 50 degrees (right triangle, at A).
Angle a is at vertex D in the right triangle ACD.

Unknowns

The measure of angle a at vertex D.

Constraints

The three angles of any triangle add to 180 degrees.
Angles ACB and ACD sit on the straight line BD, so they add to 180 degrees.

Plan

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

Subproblem 1: angle ACD is the straight-line partner of the 130-degree angle, so 180 - 130. Subproblem 2: in the right triangle ACD the three angles add to 180 degrees, so a = 180 - (CAD) - (ACD).

Execute

#7 Identify Subproblems 4.MD.C.7

Point C is on the straight line BD, so the angle on the left (ACB = 130 degrees) and the angle on the right (ACD) together make the straight angle 180 degrees.

\angle ACD = 180^\circ - 130^\circ = 50^\circ

The base is a flat 180-degree line split into the two angles at C.

#7 Identify Subproblems 4.MD.C.7

In the right-hand triangle ACD the three angles are CAD = 50 degrees, ACD = 50 degrees, and a at D. They add to 180 degrees, so subtract the two known angles.

a = 180^\circ - 50^\circ - 50^\circ = 80^\circ

Every triangle's three corners always total 180 degrees, so the third corner is the leftover.

Answer: 80 degrees

Review

Triangle ACD: 50 + 50 + 80 = 180 degrees, a valid triangle. The straight base at C: 130 + 50 = 180 degrees. Both checks hold, so a = 80 degrees.

Use the exterior-angle idea (tool 5/pattern): the 130-degree angle ACB is the outside angle of triangle ACD at C, and it equals the two far angles CAD + a, so 130 = 50 + a gives a = 80 degrees.

Standards · min grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Using the 180-degree straight line and the 180-degree triangle total to find a.
4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Reading the triangle, the base point C, and segment AC in the figure.

💡 Split the base with the 180-degree line, then the triangle's 180-degree rule hands you angle a - pure Grade 4 angle adding!

Variant 9 answer: 65 degrees

Using the fact that the three angles of a triangle add up to $180^\circ$ , find the measure of angle $a$ in the figure.

[Figure] Triangle $ABD$ has a segment $AC$ drawn from vertex $A$ down to point $C$ on the base $BD$ , splitting it into two smaller triangles. At point $C$ , the angle on the left-triangle side ( $\angle ACB$ ) is $95^\circ$ , and at vertex $A$ the angle of the right-hand triangle ( $\angle CAD$ ) is $30^\circ$ . The angle $a$ to be found is at vertex $D$ .

Show solution

Understand

Big triangle ABD has a line AC drawn from the top vertex A down to point C on the base BD, making two smaller triangles. At C, the left-triangle angle ACB is 95 degrees. At A, the right-triangle angle CAD is 30 degrees. Find angle a at vertex D.

Givens

Triangle ABD with point C on base BD and segment AC drawn.
Angle ACB = 95 degrees (left triangle, at C).
Angle CAD = 30 degrees (right triangle, at A).
Angle a is at vertex D in the right triangle ACD.

Unknowns

The measure of angle a at vertex D.

Constraints

The three angles of any triangle add to 180 degrees.
Angles ACB and ACD sit on the straight line BD, so they add to 180 degrees.

Plan

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

Subproblem 1: angle ACD is the straight-line partner of the 95-degree angle, so 180 - 95. Subproblem 2: in the right triangle ACD the three angles add to 180 degrees, so a = 180 - (CAD) - (ACD).

Execute

#7 Identify Subproblems 4.MD.C.7

Point C is on the straight line BD, so the angle on the left (ACB = 95 degrees) and the angle on the right (ACD) together make the straight angle 180 degrees.

\angle ACD = 180^\circ - 95^\circ = 85^\circ

The base is a flat 180-degree line split into the two angles at C.

#7 Identify Subproblems 4.MD.C.7

In the right-hand triangle ACD the three angles are CAD = 30 degrees, ACD = 85 degrees, and a at D. They add to 180 degrees, so subtract the two known angles.

a = 180^\circ - 30^\circ - 85^\circ = 65^\circ

Every triangle's three corners always total 180 degrees, so the third corner is the leftover.

Answer: 65 degrees

Review

Triangle ACD: 30 + 85 + 65 = 180 degrees, a valid triangle. The straight base at C: 95 + 85 = 180 degrees. Both checks hold, so a = 65 degrees.

Use the exterior-angle idea (tool 5/pattern): the 95-degree angle ACB is the outside angle of triangle ACD at C, and it equals the two far angles CAD + a, so 95 = 30 + a gives a = 65 degrees.

Standards · min grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Using the 180-degree straight line and the 180-degree triangle total to find a.
4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Reading the triangle, the base point C, and segment AC in the figure.

💡 Split the base with the 180-degree line, then the triangle's 180-degree rule hands you angle a - pure Grade 4 angle adding!

Variant 10 answer: 60 degrees

Using the fact that the three angles of a triangle add up to $180^\circ$ , find the measure of angle $a$ in the figure.

[Figure] Triangle $ABD$ has a segment $AC$ drawn from vertex $A$ down to point $C$ on the base $BD$ , splitting it into two smaller triangles. At point $C$ , the angle on the left-triangle side ( $\angle ACB$ ) is $100^\circ$ , and at vertex $A$ the angle of the right-hand triangle ( $\angle CAD$ ) is $40^\circ$ . The angle $a$ to be found is at vertex $D$ .

Show solution

Understand

Big triangle ABD has a line AC drawn from the top vertex A down to point C on the base BD, making two smaller triangles. At C, the left-triangle angle ACB is 100 degrees. At A, the right-triangle angle CAD is 40 degrees. Find angle a at vertex D.

Givens

Triangle ABD with point C on base BD and segment AC drawn.
Angle ACB = 100 degrees (left triangle, at C).
Angle CAD = 40 degrees (right triangle, at A).
Angle a is at vertex D in the right triangle ACD.

Unknowns

The measure of angle a at vertex D.

Constraints

The three angles of any triangle add to 180 degrees.
Angles ACB and ACD sit on the straight line BD, so they add to 180 degrees.

Plan

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

Subproblem 1: angle ACD is the straight-line partner of the 100-degree angle, so 180 - 100. Subproblem 2: in the right triangle ACD the three angles add to 180 degrees, so a = 180 - (CAD) - (ACD).

Execute

#7 Identify Subproblems 4.MD.C.7

Point C is on the straight line BD, so the angle on the left (ACB = 100 degrees) and the angle on the right (ACD) together make the straight angle 180 degrees.

\angle ACD = 180^\circ - 100^\circ = 80^\circ

The base is a flat 180-degree line split into the two angles at C.

#7 Identify Subproblems 4.MD.C.7

In the right-hand triangle ACD the three angles are CAD = 40 degrees, ACD = 80 degrees, and a at D. They add to 180 degrees, so subtract the two known angles.

a = 180^\circ - 40^\circ - 80^\circ = 60^\circ

Every triangle's three corners always total 180 degrees, so the third corner is the leftover.

Answer: 60 degrees

Review

Triangle ACD: 40 + 80 + 60 = 180 degrees, a valid triangle. The straight base at C: 100 + 80 = 180 degrees. Both checks hold, so a = 60 degrees.

Use the exterior-angle idea (tool 5/pattern): the 100-degree angle ACB is the outside angle of triangle ACD at C, and it equals the two far angles CAD + a, so 100 = 40 + a gives a = 60 degrees.

Standards · min grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Using the 180-degree straight line and the 180-degree triangle total to find a.
4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Reading the triangle, the base point C, and segment AC in the figure.

💡 Split the base with the 180-degree line, then the triangle's 180-degree rule hands you angle a - pure Grade 4 angle adding!