← Folded angles are equal; chain to the unknown · Transformations Preserve Measures

Folded angles are equal; chain to the unknown · 1 practice problems

4.MD.C.7

From the workbook (authentic) — 1

Real practice problems extracted and localized from the source workbook.

Workbook 1 answer: 65 degrees

A sheet of paper shaped like an equilateral triangle is folded as shown, so that the bottom-left corner is folded up onto the triangle. Find the measure of the marked angle.

A B C Q P B' 70° ?
Show solution
1 · Understandwhat's really being asked

An equilateral triangle ABC has its bottom-left corner B folded up. The crease goes from P on the base BC to Q on the left side BA, and corner B lands at B'. The folded edge PB' makes a 70-degree angle with the base. I must find the flap angle at Q, which is angle PQB'.

Givens
  • Triangle ABC is equilateral, so every interior angle is 60 degrees (in particular angle B = 60 degrees)
  • The corner B is folded over crease PQ, so B lands at B' and the fold copies every length and angle exactly
  • The folded edge PB' makes a 70-degree angle with the base toward C (angle B'PC = 70 degrees)
Unknowns
  • The measure of the flap angle at Q, angle PQB'
Constraints
  • Angles in a triangle add to 180 degrees
  • Angles on a straight line add to 180 degrees
  • Folding preserves angle sizes, so a folded angle equals the original angle it came from
2 · Planchoose the strategy

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

Folding paper is a hands-on action, so picturing the fold shows that the crease reflects corner B onto B' and keeps angles equal. Then I break the figure into small pieces: the angles meeting at P on the base, and the small triangle BPQ, and chain them to the flap angle at Q.

3 · Execute4 carry out the plan

1Use the equilateral corner

#10 Create a Physical Representation 4.MD.C.7
Because ABC is an equilateral triangle, each corner measures 60 degrees, so the angle at B is 60 degrees.
∠PBQ=60∘\angle PBQ = 60^\circ
All three corners of an equilateral triangle are the same, and three equal corners totaling 180 degrees each measure 60 degrees.

2Find the angle the crease makes with the base at P

#7 Identify Subproblems 4.MD.C.7
At P the three angles on the straight base line BC add to 180 degrees. The folded edge PB' takes up 70 degrees (angle B'PC). Folding reflects the original base part PB onto PB', so the crease PQ splits the leftover angle into two equal halves: angle BPQ = angle B'PQ. The leftover is 180 - 70 = 110 degrees, so each half is 110 / 2 = 55 degrees.
∠BPQ=180∘−70∘2=110∘2=55∘\angle BPQ = \dfrac{180^\circ - 70^\circ}{2} = \dfrac{110^\circ}{2} = 55^\circ
The fold makes a mirror image, so the crease sits exactly in the middle of the angle between the original edge and the folded edge.

3Find the flap angle in triangle BPQ

#7 Identify Subproblems 4.MD.C.7
In triangle BPQ the three angles add to 180 degrees. With angle B = 60 degrees and angle BPQ = 55 degrees, the angle at Q is angle BQP = 180 - 60 - 55 = 65 degrees.
∠BQP=180∘−60∘−55∘=65∘\angle BQP = 180^\circ - 60^\circ - 55^\circ = 65^\circ
Once two angles of a triangle are known, the third is whatever is left over from 180 degrees.

4Carry the angle through the fold to B'

#10 Create a Physical Representation 4.MD.C.7
Folding does not change angle sizes, so the flap angle at Q after folding equals the angle before folding: angle PQB' = angle PQB = 65 degrees.
∠PQB′=∠BQP=65∘\angle PQB' = \angle BQP = 65^\circ
A folded shape lands exactly on top of its original, so the matching angle keeps the same size.
Answer: 65 degrees
4 · Reviewdoes it hold up?

The flap angle 65 degrees is acute, which matches the narrow corner shown at Q. Checking triangle BPQ: 60 + 55 + 65 = 180 degrees, and the folded corner at B' keeps its 60 degrees (180 - 55 - 65 = 60), exactly the equilateral corner that was folded. Everything is consistent.

Another way: Draw the figure to scale (tool 1) and measure the flap angle at Q with a protractor to confirm the 65-degree answer found by angle chasing.

Standardsmin grade 4
  • 4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Splitting the straight-line angle at P, using the equilateral 60-degree corner, and chaining through the triangle-sum and the equal folded angle to reach the flap angle at Q.
💡Takeaway. Split the straight line in half at the fold, then use a triangle's three angles adding to 180 degrees -- only Grade 4 angle work needed!