Exercise 34 Page 427

We are asked to prove Theorem 6.14. We need to show that if the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. Let's copy the diagram and label some angles and the intersection point of the two diagonals.

We are given two pieces of information about quadrilateral WXYZ.

We know that WXYZ is a parallelogram, so according to Theorem 6.7 its diagonals bisect each other.
We are also given that the diagonals are congruent. Along with the previous observation, this means that the four segments connecting their intersection point with the vertices are congruent.

We can now conclude that the triangles formed by the diagonals and the sides of the parallelogram are isosceles. This implies the congruence of two pairs of the labeled angles.

Congruent Angles	Reason
∠ 1≅ ∠ 2	△ WOX is isosceles.
∠ 3≅ ∠ 4	△ XOY is isosceles.

We can now use the Triangle Angle-Sum Theorem in △ WXY to find the measure of the angle at X.

m∠ 1+(m∠ 2+m∠ 3)+m∠ 4=180

SubstituteII

m∠ 1= m∠ 2, m∠ 4= m∠ 3

m∠ 2+m∠ 2+m∠ 3+ m∠ 3=180

AddTerms

Add terms

2m∠ 2+2m∠ 3=180

DivEqn

.LHS /2.=.RHS /2.

m∠ 2+m∠ 3=90

Since ∠ 2 and ∠ 3 together form ∠ X, this means that the parallelogram has a right angle at vertex X. According to Theorem 6.6 this means that the parallelogram is a rectangle. We can summarize the process above in a two-column proof.

Completed Proof

2 &Given:&& WXYZ is a parallelogram & && WY≅YZ &Prove:&& WXYZ is a rectangle Proof:

Statements	Reasons
1. WXYZ is a parallelogram.	1. Given
2. WO≅YO	2. XZ bisects WY (Theorem 6.7)
3. XO≅ZO	3. WY bisects XZ (Theorem 6.7)
4. WY≅XZ	4. Given
5. WO≅XO≅YO≅ZO	5. Half of congruent segments
6. ∠ XWO≅∠ WXO	6. Isosceles Triangle Theorem
7. ∠ YXO≅∠ XYO	7. Isosceles Triangle Theorem
8. m∠ X=90	8. Triangle Angle-Sum Theorem
9. WXYZ is a rectangle.	9. Theorem 6.6

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Completed Proof