Practice · Folding makes congruent corresponding angles · Sensim Math

Variant 1 easy

A rectangular sheet of paper is folded as shown in the figure. Find the measure of angle ⓐ.

Figure description: One corner of a rectangular sheet of paper has been folded up once. The fold creates a point M that sticks up at the top, and the fold line meets the bottom edge of the paper at point N. At N (the left end of the bottom edge), the angle between the folded edge and the bottom edge measures $18^\circ$ . The angle you must find, ⓐ, is the angle marked near the top point M, where the two folded faces meet.

Your answer

Show solution

1 · Understandwhat's really being asked

A rectangular sheet of paper has one corner folded up. The fold line meets the bottom edge at point N, and the folded flap rises to a peak M at the top. At N the angle between the folded edge and the bottom edge is 18 deg. I need angle a at the peak M, where the two folded faces meet.

Givens

A rectangular sheet is folded once at a corner.
The fold line meets the bottom edge at N.
The peak of the folded flap is M.
At N the angle between the folded edge and the bottom edge is 18 deg.
Folding is a reflection, so it preserves lengths and angles.

Unknowns

The measure of angle a at the peak M.

Constraints

Folding reflects the paper, making the two folded faces mirror images.
The standing flap is an isosceles triangle whose base angles are equal.

2 · Planchoose the strategy

#10 Create a Physical Representation · also uses: #1 Draw a Diagram#7 Identify Subproblems

Treat the fold as a real reflection. The flap that rises to M is the mirror image of part of the paper, so the two faces meeting at M are equal and form an isosceles triangle. The slant at the base appears on both sides of the fold, and the peak angle is what is left of the triangle's 180 deg after removing those two equal base pieces.

3 · Execute3 carry out the plan

1The fold makes two equal base angles

#10 Create a Physical Representation 4.G.A.1

Folding reflects the corner across the crease, so the two folded edges make the same 18 deg slant with the bottom edge. The standing flap is therefore an isosceles triangle with two equal base angles of 18 deg.

\text{both base angles} = 18^\circ \text{ each}

A fold is a mirror: whatever angle the paper makes on one side of the crease, it makes the same angle on the other side.

2Use the triangle angle sum

#7 Identify Subproblems 4.MD.C.7

The three angles of the flap triangle add to 180 deg. The two equal base angles of 18 deg each leave the peak angle a as the remainder.

a = 180^\circ - 18^\circ - 18^\circ

The folded flap is an isosceles triangle with two equal base angles of 18 deg, so the top angle is the rest of the 180 deg total.

3Compute the peak angle

#7 Identify Subproblems 4.MD.C.7

Subtract the two equal base angles from 180 deg.

a = 180^\circ - 36^\circ = 144^\circ

Two 18 deg corners leave a 144 deg angle at the top of the folded flap.

Answer: 144 degrees

4 · Reviewdoes it hold up?

Check: 18 + 18 + 144 = 180 deg, exactly a triangle's angle sum, so the peak angle 144 deg is consistent with the two equal 18 deg base angles.

Another way: Use Draw a Diagram (tool 1): mark the equal fold angles as 18 deg each in the isosceles triangle at the peak, then apply the triangle angle sum 180 - 2(18) = 144 deg.

Standardsmin grade 4

4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Identifying the equal angles created by the fold (reflection) at the base.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting the two 18 deg base angles from 180 deg to find the peak angle.

💡Takeaway. A fold is a mirror, so the two base angles match at 18 deg each; the peak is just 180 deg minus those two!

Variant 2 easy

A rectangular sheet of paper is folded as shown in the figure. Find the measure of angle ⓐ.

Figure description: One corner of a rectangular sheet of paper has been folded up once. The fold creates a point M that sticks up at the top, and the fold line meets the bottom edge of the paper at point N. At N (the left end of the bottom edge), the angle between the folded edge and the bottom edge measures $20^\circ$ . The angle you must find, ⓐ, is the angle marked near the top point M, where the two folded faces meet.

Your answer

Show solution

1 · Understandwhat's really being asked

A rectangular sheet of paper has one corner folded up. The fold line meets the bottom edge at point N, and the folded flap rises to a peak M at the top. At N the angle between the folded edge and the bottom edge is 20 deg. I need angle a at the peak M, where the two folded faces meet.

Givens

A rectangular sheet is folded once at a corner.
The fold line meets the bottom edge at N.
The peak of the folded flap is M.
At N the angle between the folded edge and the bottom edge is 20 deg.
Folding is a reflection, so it preserves lengths and angles.

Unknowns

The measure of angle a at the peak M.

Constraints

Folding reflects the paper, making the two folded faces mirror images.
The standing flap is an isosceles triangle whose base angles are equal.

2 · Planchoose the strategy

#10 Create a Physical Representation · also uses: #1 Draw a Diagram#7 Identify Subproblems

Treat the fold as a real reflection. The flap that rises to M is the mirror image of part of the paper, so the two faces meeting at M are equal and form an isosceles triangle. The slant at the base appears on both sides of the fold, and the peak angle is what is left of the triangle's 180 deg after removing those two equal base pieces.

3 · Execute3 carry out the plan

1The fold makes two equal base angles

#10 Create a Physical Representation 4.G.A.1

Folding reflects the corner across the crease, so the two folded edges make the same 20 deg slant with the bottom edge. The standing flap is therefore an isosceles triangle with two equal base angles of 20 deg.

\text{both base angles} = 20^\circ \text{ each}

A fold is a mirror: whatever angle the paper makes on one side of the crease, it makes the same angle on the other side.

2Use the triangle angle sum

#7 Identify Subproblems 4.MD.C.7

The three angles of the flap triangle add to 180 deg. The two equal base angles of 20 deg each leave the peak angle a as the remainder.

a = 180^\circ - 20^\circ - 20^\circ

The folded flap is an isosceles triangle with two equal base angles of 20 deg, so the top angle is the rest of the 180 deg total.

3Compute the peak angle

#7 Identify Subproblems 4.MD.C.7

Subtract the two equal base angles from 180 deg.

a = 180^\circ - 40^\circ = 140^\circ

Two 20 deg corners leave a 140 deg angle at the top of the folded flap.

Answer: 140 degrees

4 · Reviewdoes it hold up?

Check: 20 + 20 + 140 = 180 deg, exactly a triangle's angle sum, so the peak angle 140 deg is consistent with the two equal 20 deg base angles.

Another way: Use Draw a Diagram (tool 1): mark the equal fold angles as 20 deg each in the isosceles triangle at the peak, then apply the triangle angle sum 180 - 2(20) = 140 deg.

Standardsmin grade 4

4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Identifying the equal angles created by the fold (reflection) at the base.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting the two 20 deg base angles from 180 deg to find the peak angle.

💡Takeaway. A fold is a mirror, so the two base angles match at 20 deg each; the peak is just 180 deg minus those two!

Variant 3 easy

A rectangular sheet of paper is folded as shown in the figure. Find the measure of angle ⓐ.

Figure description: One corner of a rectangular sheet of paper has been folded up once. The fold creates a point M that sticks up at the top, and the fold line meets the bottom edge of the paper at point N. At N (the left end of the bottom edge), the angle between the folded edge and the bottom edge measures $25^\circ$ . The angle you must find, ⓐ, is the angle marked near the top point M, where the two folded faces meet.

Your answer

Show solution

1 · Understandwhat's really being asked

A rectangular sheet of paper has one corner folded up. The fold line meets the bottom edge at point N, and the folded flap rises to a peak M at the top. At N the angle between the folded edge and the bottom edge is 25 deg. I need angle a at the peak M, where the two folded faces meet.

Givens

A rectangular sheet is folded once at a corner.
The fold line meets the bottom edge at N.
The peak of the folded flap is M.
At N the angle between the folded edge and the bottom edge is 25 deg.
Folding is a reflection, so it preserves lengths and angles.

Unknowns

The measure of angle a at the peak M.

Constraints

Folding reflects the paper, making the two folded faces mirror images.
The standing flap is an isosceles triangle whose base angles are equal.

2 · Planchoose the strategy

#10 Create a Physical Representation · also uses: #1 Draw a Diagram#7 Identify Subproblems

Treat the fold as a real reflection. The flap that rises to M is the mirror image of part of the paper, so the two faces meeting at M are equal and form an isosceles triangle. The slant at the base appears on both sides of the fold, and the peak angle is what is left of the triangle's 180 deg after removing those two equal base pieces.

3 · Execute3 carry out the plan

1The fold makes two equal base angles

#10 Create a Physical Representation 4.G.A.1

Folding reflects the corner across the crease, so the two folded edges make the same 25 deg slant with the bottom edge. The standing flap is therefore an isosceles triangle with two equal base angles of 25 deg.

\text{both base angles} = 25^\circ \text{ each}

A fold is a mirror: whatever angle the paper makes on one side of the crease, it makes the same angle on the other side.

2Use the triangle angle sum

#7 Identify Subproblems 4.MD.C.7

The three angles of the flap triangle add to 180 deg. The two equal base angles of 25 deg each leave the peak angle a as the remainder.

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

The folded flap is an isosceles triangle with two equal base angles of 25 deg, so the top angle is the rest of the 180 deg total.

3Compute the peak angle

#7 Identify Subproblems 4.MD.C.7

Subtract the two equal base angles from 180 deg.

a = 180^\circ - 50^\circ = 130^\circ

Two 25 deg corners leave a 130 deg angle at the top of the folded flap.

Answer: 130 degrees

4 · Reviewdoes it hold up?

Check: 25 + 25 + 130 = 180 deg, exactly a triangle's angle sum, so the peak angle 130 deg is consistent with the two equal 25 deg base angles.

Another way: Use Draw a Diagram (tool 1): mark the equal fold angles as 25 deg each in the isosceles triangle at the peak, then apply the triangle angle sum 180 - 2(25) = 130 deg.

Standardsmin grade 4

4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Identifying the equal angles created by the fold (reflection) at the base.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting the two 25 deg base angles from 180 deg to find the peak angle.

💡Takeaway. A fold is a mirror, so the two base angles match at 25 deg each; the peak is just 180 deg minus those two!

Variant 4 medium

A rectangular sheet of paper is folded as shown in the figure. Find the measure of angle ⓐ.

Figure description: One corner of a rectangular sheet of paper has been folded up once. The fold creates a point M that sticks up at the top, and the fold line meets the bottom edge of the paper at point N. At N (the left end of the bottom edge), the angle between the folded edge and the bottom edge measures $28^\circ$ . The angle you must find, ⓐ, is the angle marked near the top point M, where the two folded faces meet.

Your answer

Show solution

1 · Understandwhat's really being asked

A rectangular sheet of paper has one corner folded up. The fold line meets the bottom edge at point N, and the folded flap rises to a peak M at the top. At N the angle between the folded edge and the bottom edge is 28 deg. I need angle a at the peak M, where the two folded faces meet.

Givens

A rectangular sheet is folded once at a corner.
The fold line meets the bottom edge at N.
The peak of the folded flap is M.
At N the angle between the folded edge and the bottom edge is 28 deg.
Folding is a reflection, so it preserves lengths and angles.

Unknowns

The measure of angle a at the peak M.

Constraints

Folding reflects the paper, making the two folded faces mirror images.
The standing flap is an isosceles triangle whose base angles are equal.

2 · Planchoose the strategy

#10 Create a Physical Representation · also uses: #1 Draw a Diagram#7 Identify Subproblems

Treat the fold as a real reflection. The flap that rises to M is the mirror image of part of the paper, so the two faces meeting at M are equal and form an isosceles triangle. The slant at the base appears on both sides of the fold, and the peak angle is what is left of the triangle's 180 deg after removing those two equal base pieces.

3 · Execute3 carry out the plan

1The fold makes two equal base angles

#10 Create a Physical Representation 4.G.A.1

Folding reflects the corner across the crease, so the two folded edges make the same 28 deg slant with the bottom edge. The standing flap is therefore an isosceles triangle with two equal base angles of 28 deg.

\text{both base angles} = 28^\circ \text{ each}

A fold is a mirror: whatever angle the paper makes on one side of the crease, it makes the same angle on the other side.

2Use the triangle angle sum

#7 Identify Subproblems 4.MD.C.7

The three angles of the flap triangle add to 180 deg. The two equal base angles of 28 deg each leave the peak angle a as the remainder.

a = 180^\circ - 28^\circ - 28^\circ

The folded flap is an isosceles triangle with two equal base angles of 28 deg, so the top angle is the rest of the 180 deg total.

3Compute the peak angle

#7 Identify Subproblems 4.MD.C.7

Subtract the two equal base angles from 180 deg.

a = 180^\circ - 56^\circ = 124^\circ

Two 28 deg corners leave a 124 deg angle at the top of the folded flap.

Answer: 124 degrees

4 · Reviewdoes it hold up?

Check: 28 + 28 + 124 = 180 deg, exactly a triangle's angle sum, so the peak angle 124 deg is consistent with the two equal 28 deg base angles.

Another way: Use Draw a Diagram (tool 1): mark the equal fold angles as 28 deg each in the isosceles triangle at the peak, then apply the triangle angle sum 180 - 2(28) = 124 deg.

Standardsmin grade 4

4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Identifying the equal angles created by the fold (reflection) at the base.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting the two 28 deg base angles from 180 deg to find the peak angle.

💡Takeaway. A fold is a mirror, so the two base angles match at 28 deg each; the peak is just 180 deg minus those two!

Variant 5 medium

A rectangular sheet of paper is folded as shown in the figure. Find the measure of angle ⓐ.

Figure description: One corner of a rectangular sheet of paper has been folded up once. The fold creates a point M that sticks up at the top, and the fold line meets the bottom edge of the paper at point N. At N (the left end of the bottom edge), the angle between the folded edge and the bottom edge measures $30^\circ$ . The angle you must find, ⓐ, is the angle marked near the top point M, where the two folded faces meet.

Your answer

Show solution

1 · Understandwhat's really being asked

A rectangular sheet of paper has one corner folded up. The fold line meets the bottom edge at point N, and the folded flap rises to a peak M at the top. At N the angle between the folded edge and the bottom edge is 30 deg. I need angle a at the peak M, where the two folded faces meet.

Givens

A rectangular sheet is folded once at a corner.
The fold line meets the bottom edge at N.
The peak of the folded flap is M.
At N the angle between the folded edge and the bottom edge is 30 deg.
Folding is a reflection, so it preserves lengths and angles.

Unknowns

The measure of angle a at the peak M.

Constraints

Folding reflects the paper, making the two folded faces mirror images.
The standing flap is an isosceles triangle whose base angles are equal.

2 · Planchoose the strategy

#10 Create a Physical Representation · also uses: #1 Draw a Diagram#7 Identify Subproblems

Treat the fold as a real reflection. The flap that rises to M is the mirror image of part of the paper, so the two faces meeting at M are equal and form an isosceles triangle. The slant at the base appears on both sides of the fold, and the peak angle is what is left of the triangle's 180 deg after removing those two equal base pieces.

3 · Execute3 carry out the plan

1The fold makes two equal base angles

#10 Create a Physical Representation 4.G.A.1

Folding reflects the corner across the crease, so the two folded edges make the same 30 deg slant with the bottom edge. The standing flap is therefore an isosceles triangle with two equal base angles of 30 deg.

\text{both base angles} = 30^\circ \text{ each}

A fold is a mirror: whatever angle the paper makes on one side of the crease, it makes the same angle on the other side.

2Use the triangle angle sum

#7 Identify Subproblems 4.MD.C.7

The three angles of the flap triangle add to 180 deg. The two equal base angles of 30 deg each leave the peak angle a as the remainder.

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

The folded flap is an isosceles triangle with two equal base angles of 30 deg, so the top angle is the rest of the 180 deg total.

3Compute the peak angle

#7 Identify Subproblems 4.MD.C.7

Subtract the two equal base angles from 180 deg.

a = 180^\circ - 60^\circ = 120^\circ

Two 30 deg corners leave a 120 deg angle at the top of the folded flap.

Answer: 120 degrees

4 · Reviewdoes it hold up?

Check: 30 + 30 + 120 = 180 deg, exactly a triangle's angle sum, so the peak angle 120 deg is consistent with the two equal 30 deg base angles.

Another way: Use Draw a Diagram (tool 1): mark the equal fold angles as 30 deg each in the isosceles triangle at the peak, then apply the triangle angle sum 180 - 2(30) = 120 deg.

Standardsmin grade 4

4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Identifying the equal angles created by the fold (reflection) at the base.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting the two 30 deg base angles from 180 deg to find the peak angle.

💡Takeaway. A fold is a mirror, so the two base angles match at 30 deg each; the peak is just 180 deg minus those two!

Variant 6 medium

A rectangular sheet of paper is folded as shown in the figure. Find the measure of angle ⓐ.

Figure description: One corner of a rectangular sheet of paper has been folded up once. The fold creates a point M that sticks up at the top, and the fold line meets the bottom edge of the paper at point N. At N (the left end of the bottom edge), the angle between the folded edge and the bottom edge measures $32^\circ$ . The angle you must find, ⓐ, is the angle marked near the top point M, where the two folded faces meet.

Your answer

Show solution

1 · Understandwhat's really being asked

A rectangular sheet of paper has one corner folded up. The fold line meets the bottom edge at point N, and the folded flap rises to a peak M at the top. At N the angle between the folded edge and the bottom edge is 32 deg. I need angle a at the peak M, where the two folded faces meet.

Givens

A rectangular sheet is folded once at a corner.
The fold line meets the bottom edge at N.
The peak of the folded flap is M.
At N the angle between the folded edge and the bottom edge is 32 deg.
Folding is a reflection, so it preserves lengths and angles.

Unknowns

The measure of angle a at the peak M.

Constraints

Folding reflects the paper, making the two folded faces mirror images.
The standing flap is an isosceles triangle whose base angles are equal.

2 · Planchoose the strategy

#10 Create a Physical Representation · also uses: #1 Draw a Diagram#7 Identify Subproblems

Treat the fold as a real reflection. The flap that rises to M is the mirror image of part of the paper, so the two faces meeting at M are equal and form an isosceles triangle. The slant at the base appears on both sides of the fold, and the peak angle is what is left of the triangle's 180 deg after removing those two equal base pieces.

3 · Execute3 carry out the plan

1The fold makes two equal base angles

#10 Create a Physical Representation 4.G.A.1

Folding reflects the corner across the crease, so the two folded edges make the same 32 deg slant with the bottom edge. The standing flap is therefore an isosceles triangle with two equal base angles of 32 deg.

\text{both base angles} = 32^\circ \text{ each}

A fold is a mirror: whatever angle the paper makes on one side of the crease, it makes the same angle on the other side.

2Use the triangle angle sum

#7 Identify Subproblems 4.MD.C.7

The three angles of the flap triangle add to 180 deg. The two equal base angles of 32 deg each leave the peak angle a as the remainder.

a = 180^\circ - 32^\circ - 32^\circ

The folded flap is an isosceles triangle with two equal base angles of 32 deg, so the top angle is the rest of the 180 deg total.

3Compute the peak angle

#7 Identify Subproblems 4.MD.C.7

Subtract the two equal base angles from 180 deg.

a = 180^\circ - 64^\circ = 116^\circ

Two 32 deg corners leave a 116 deg angle at the top of the folded flap.

Answer: 116 degrees

4 · Reviewdoes it hold up?

Check: 32 + 32 + 116 = 180 deg, exactly a triangle's angle sum, so the peak angle 116 deg is consistent with the two equal 32 deg base angles.

Another way: Use Draw a Diagram (tool 1): mark the equal fold angles as 32 deg each in the isosceles triangle at the peak, then apply the triangle angle sum 180 - 2(32) = 116 deg.

Standardsmin grade 4

4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Identifying the equal angles created by the fold (reflection) at the base.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting the two 32 deg base angles from 180 deg to find the peak angle.

💡Takeaway. A fold is a mirror, so the two base angles match at 32 deg each; the peak is just 180 deg minus those two!

Variant 7 medium

A rectangular sheet of paper is folded as shown in the figure. Find the measure of angle ⓐ.

Figure description: One corner of a rectangular sheet of paper has been folded up once. The fold creates a point M that sticks up at the top, and the fold line meets the bottom edge of the paper at point N. At N (the left end of the bottom edge), the angle between the folded edge and the bottom edge measures $35^\circ$ . The angle you must find, ⓐ, is the angle marked near the top point M, where the two folded faces meet.

Your answer

Show solution

1 · Understandwhat's really being asked

A rectangular sheet of paper has one corner folded up. The fold line meets the bottom edge at point N, and the folded flap rises to a peak M at the top. At N the angle between the folded edge and the bottom edge is 35 deg. I need angle a at the peak M, where the two folded faces meet.

Givens

A rectangular sheet is folded once at a corner.
The fold line meets the bottom edge at N.
The peak of the folded flap is M.
At N the angle between the folded edge and the bottom edge is 35 deg.
Folding is a reflection, so it preserves lengths and angles.

Unknowns

The measure of angle a at the peak M.

Constraints

Folding reflects the paper, making the two folded faces mirror images.
The standing flap is an isosceles triangle whose base angles are equal.

2 · Planchoose the strategy

#10 Create a Physical Representation · also uses: #1 Draw a Diagram#7 Identify Subproblems

Treat the fold as a real reflection. The flap that rises to M is the mirror image of part of the paper, so the two faces meeting at M are equal and form an isosceles triangle. The slant at the base appears on both sides of the fold, and the peak angle is what is left of the triangle's 180 deg after removing those two equal base pieces.

3 · Execute3 carry out the plan

1The fold makes two equal base angles

#10 Create a Physical Representation 4.G.A.1

Folding reflects the corner across the crease, so the two folded edges make the same 35 deg slant with the bottom edge. The standing flap is therefore an isosceles triangle with two equal base angles of 35 deg.

\text{both base angles} = 35^\circ \text{ each}

A fold is a mirror: whatever angle the paper makes on one side of the crease, it makes the same angle on the other side.

2Use the triangle angle sum

#7 Identify Subproblems 4.MD.C.7

The three angles of the flap triangle add to 180 deg. The two equal base angles of 35 deg each leave the peak angle a as the remainder.

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

The folded flap is an isosceles triangle with two equal base angles of 35 deg, so the top angle is the rest of the 180 deg total.

3Compute the peak angle

#7 Identify Subproblems 4.MD.C.7

Subtract the two equal base angles from 180 deg.

a = 180^\circ - 70^\circ = 110^\circ

Two 35 deg corners leave a 110 deg angle at the top of the folded flap.

Answer: 110 degrees

4 · Reviewdoes it hold up?

Check: 35 + 35 + 110 = 180 deg, exactly a triangle's angle sum, so the peak angle 110 deg is consistent with the two equal 35 deg base angles.

Another way: Use Draw a Diagram (tool 1): mark the equal fold angles as 35 deg each in the isosceles triangle at the peak, then apply the triangle angle sum 180 - 2(35) = 110 deg.

Standardsmin grade 4

4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Identifying the equal angles created by the fold (reflection) at the base.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting the two 35 deg base angles from 180 deg to find the peak angle.

💡Takeaway. A fold is a mirror, so the two base angles match at 35 deg each; the peak is just 180 deg minus those two!

Variant 8 hard

A rectangular sheet of paper is folded as shown in the figure. Find the measure of angle ⓐ.

Figure description: One corner of a rectangular sheet of paper has been folded up once. The fold creates a point M that sticks up at the top, and the fold line meets the bottom edge of the paper at point N. At N (the left end of the bottom edge), the angle between the folded edge and the bottom edge measures $38^\circ$ . The angle you must find, ⓐ, is the angle marked near the top point M, where the two folded faces meet.

Your answer

Show solution

1 · Understandwhat's really being asked

A rectangular sheet of paper has one corner folded up. The fold line meets the bottom edge at point N, and the folded flap rises to a peak M at the top. At N the angle between the folded edge and the bottom edge is 38 deg. I need angle a at the peak M, where the two folded faces meet.

Givens

A rectangular sheet is folded once at a corner.
The fold line meets the bottom edge at N.
The peak of the folded flap is M.
At N the angle between the folded edge and the bottom edge is 38 deg.
Folding is a reflection, so it preserves lengths and angles.

Unknowns

The measure of angle a at the peak M.

Constraints

Folding reflects the paper, making the two folded faces mirror images.
The standing flap is an isosceles triangle whose base angles are equal.

2 · Planchoose the strategy

#10 Create a Physical Representation · also uses: #1 Draw a Diagram#7 Identify Subproblems

Treat the fold as a real reflection. The flap that rises to M is the mirror image of part of the paper, so the two faces meeting at M are equal and form an isosceles triangle. The slant at the base appears on both sides of the fold, and the peak angle is what is left of the triangle's 180 deg after removing those two equal base pieces.

3 · Execute3 carry out the plan

1The fold makes two equal base angles

#10 Create a Physical Representation 4.G.A.1

Folding reflects the corner across the crease, so the two folded edges make the same 38 deg slant with the bottom edge. The standing flap is therefore an isosceles triangle with two equal base angles of 38 deg.

\text{both base angles} = 38^\circ \text{ each}

A fold is a mirror: whatever angle the paper makes on one side of the crease, it makes the same angle on the other side.

2Use the triangle angle sum

#7 Identify Subproblems 4.MD.C.7

The three angles of the flap triangle add to 180 deg. The two equal base angles of 38 deg each leave the peak angle a as the remainder.

a = 180^\circ - 38^\circ - 38^\circ

The folded flap is an isosceles triangle with two equal base angles of 38 deg, so the top angle is the rest of the 180 deg total.

3Compute the peak angle

#7 Identify Subproblems 4.MD.C.7

Subtract the two equal base angles from 180 deg.

a = 180^\circ - 76^\circ = 104^\circ

Two 38 deg corners leave a 104 deg angle at the top of the folded flap.

Answer: 104 degrees

4 · Reviewdoes it hold up?

Check: 38 + 38 + 104 = 180 deg, exactly a triangle's angle sum, so the peak angle 104 deg is consistent with the two equal 38 deg base angles.

Another way: Use Draw a Diagram (tool 1): mark the equal fold angles as 38 deg each in the isosceles triangle at the peak, then apply the triangle angle sum 180 - 2(38) = 104 deg.

Standardsmin grade 4

4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Identifying the equal angles created by the fold (reflection) at the base.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting the two 38 deg base angles from 180 deg to find the peak angle.

💡Takeaway. A fold is a mirror, so the two base angles match at 38 deg each; the peak is just 180 deg minus those two!

Variant 9 hard

A rectangular sheet of paper is folded as shown in the figure. Find the measure of angle ⓐ.

Figure description: One corner of a rectangular sheet of paper has been folded up once. The fold creates a point M that sticks up at the top, and the fold line meets the bottom edge of the paper at point N. At N (the left end of the bottom edge), the angle between the folded edge and the bottom edge measures $40^\circ$ . The angle you must find, ⓐ, is the angle marked near the top point M, where the two folded faces meet.

Your answer

Show solution

1 · Understandwhat's really being asked

A rectangular sheet of paper has one corner folded up. The fold line meets the bottom edge at point N, and the folded flap rises to a peak M at the top. At N the angle between the folded edge and the bottom edge is 40 deg. I need angle a at the peak M, where the two folded faces meet.

Givens

A rectangular sheet is folded once at a corner.
The fold line meets the bottom edge at N.
The peak of the folded flap is M.
At N the angle between the folded edge and the bottom edge is 40 deg.
Folding is a reflection, so it preserves lengths and angles.

Unknowns

The measure of angle a at the peak M.

Constraints

Folding reflects the paper, making the two folded faces mirror images.
The standing flap is an isosceles triangle whose base angles are equal.

2 · Planchoose the strategy

#10 Create a Physical Representation · also uses: #1 Draw a Diagram#7 Identify Subproblems

Treat the fold as a real reflection. The flap that rises to M is the mirror image of part of the paper, so the two faces meeting at M are equal and form an isosceles triangle. The slant at the base appears on both sides of the fold, and the peak angle is what is left of the triangle's 180 deg after removing those two equal base pieces.

3 · Execute3 carry out the plan

1The fold makes two equal base angles

#10 Create a Physical Representation 4.G.A.1

Folding reflects the corner across the crease, so the two folded edges make the same 40 deg slant with the bottom edge. The standing flap is therefore an isosceles triangle with two equal base angles of 40 deg.

\text{both base angles} = 40^\circ \text{ each}

A fold is a mirror: whatever angle the paper makes on one side of the crease, it makes the same angle on the other side.

2Use the triangle angle sum

#7 Identify Subproblems 4.MD.C.7

The three angles of the flap triangle add to 180 deg. The two equal base angles of 40 deg each leave the peak angle a as the remainder.

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

The folded flap is an isosceles triangle with two equal base angles of 40 deg, so the top angle is the rest of the 180 deg total.

3Compute the peak angle

#7 Identify Subproblems 4.MD.C.7

Subtract the two equal base angles from 180 deg.

a = 180^\circ - 80^\circ = 100^\circ

Two 40 deg corners leave a 100 deg angle at the top of the folded flap.

Answer: 100 degrees

4 · Reviewdoes it hold up?

Check: 40 + 40 + 100 = 180 deg, exactly a triangle's angle sum, so the peak angle 100 deg is consistent with the two equal 40 deg base angles.

Another way: Use Draw a Diagram (tool 1): mark the equal fold angles as 40 deg each in the isosceles triangle at the peak, then apply the triangle angle sum 180 - 2(40) = 100 deg.

Standardsmin grade 4

4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Identifying the equal angles created by the fold (reflection) at the base.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting the two 40 deg base angles from 180 deg to find the peak angle.

💡Takeaway. A fold is a mirror, so the two base angles match at 40 deg each; the peak is just 180 deg minus those two!

Variant 10 hard

A rectangular sheet of paper is folded as shown in the figure. Find the measure of angle ⓐ.

Figure description: One corner of a rectangular sheet of paper has been folded up once. The fold creates a point M that sticks up at the top, and the fold line meets the bottom edge of the paper at point N. At N (the left end of the bottom edge), the angle between the folded edge and the bottom edge measures $45^\circ$ . The angle you must find, ⓐ, is the angle marked near the top point M, where the two folded faces meet.

Your answer

Show solution

1 · Understandwhat's really being asked

A rectangular sheet of paper has one corner folded up. The fold line meets the bottom edge at point N, and the folded flap rises to a peak M at the top. At N the angle between the folded edge and the bottom edge is 45 deg. I need angle a at the peak M, where the two folded faces meet.

Givens

A rectangular sheet is folded once at a corner.
The fold line meets the bottom edge at N.
The peak of the folded flap is M.
At N the angle between the folded edge and the bottom edge is 45 deg.
Folding is a reflection, so it preserves lengths and angles.

Unknowns

The measure of angle a at the peak M.

Constraints

Folding reflects the paper, making the two folded faces mirror images.
The standing flap is an isosceles triangle whose base angles are equal.

2 · Planchoose the strategy

#10 Create a Physical Representation · also uses: #1 Draw a Diagram#7 Identify Subproblems

Treat the fold as a real reflection. The flap that rises to M is the mirror image of part of the paper, so the two faces meeting at M are equal and form an isosceles triangle. The slant at the base appears on both sides of the fold, and the peak angle is what is left of the triangle's 180 deg after removing those two equal base pieces.

3 · Execute3 carry out the plan

1The fold makes two equal base angles

#10 Create a Physical Representation 4.G.A.1

Folding reflects the corner across the crease, so the two folded edges make the same 45 deg slant with the bottom edge. The standing flap is therefore an isosceles triangle with two equal base angles of 45 deg.

\text{both base angles} = 45^\circ \text{ each}

A fold is a mirror: whatever angle the paper makes on one side of the crease, it makes the same angle on the other side.

2Use the triangle angle sum

#7 Identify Subproblems 4.MD.C.7

The three angles of the flap triangle add to 180 deg. The two equal base angles of 45 deg each leave the peak angle a as the remainder.

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

The folded flap is an isosceles triangle with two equal base angles of 45 deg, so the top angle is the rest of the 180 deg total.

3Compute the peak angle

#7 Identify Subproblems 4.MD.C.7

Subtract the two equal base angles from 180 deg.

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

Two 45 deg corners leave a 90 deg angle at the top of the folded flap.

Answer: 90 degrees

4 · Reviewdoes it hold up?

Check: 45 + 45 + 90 = 180 deg, exactly a triangle's angle sum, so the peak angle 90 deg is consistent with the two equal 45 deg base angles.

Another way: Use Draw a Diagram (tool 1): mark the equal fold angles as 45 deg each in the isosceles triangle at the peak, then apply the triangle angle sum 180 - 2(45) = 90 deg.

Standardsmin grade 4

4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Identifying the equal angles created by the fold (reflection) at the base.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting the two 45 deg base angles from 180 deg to find the peak angle.

💡Takeaway. A fold is a mirror, so the two base angles match at 45 deg each; the peak is just 180 deg minus those two!

Folding makes congruent corresponding angles · 10 practice problems

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