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Chapter 5: Q32 (page 188)

Prove Theorem 5-12.

Short Answer

Expert verified

Theorem 5-12 is proved by substitution, property, segment addition postulate.

Step by step solution

01

Step 1. State the theorem 5-12.

Diagonals of the rectangle are congruent for rectangle QRST , it is to be shown that QS¯≅RT¯.

02

Step 2. State the proof.

The given statement can be proved as below:

Statement

Reasons

Rectangle QRST.

Given.

is the midpoint RT¯ and QS¯.

Theorem 5-3 diagonals of a parallelogram bisect each other.

In right ΔTQR, O is the midpoint RT¯.

Proved above.

OT¯≅OQ¯≅OR¯.

Theorem 5-15 the midpoint of the hypotenuse of a right triangle is equidistant from the three vertices.

In right ΔSRQ, O is the midpoint QS¯.

Proved above.

OS¯≅OQ¯≅OR¯.

Theorem 5-15 the midpoint of the hypotenuse of a right triangle is equidistant from the three vertices.

OS¯≅OT¯≅OQ¯≅OR¯.

Substitution property.

OS¯≅OQ¯≅OR¯RT¯=OT¯+OR¯

Segment addition postulate.

QS¯=2OQ¯RT¯=2OQ¯

Substitution property.

QS¯=RT¯

Substitution Property.

03

Step 3. State the conclusion.

Therefore, Theorem 5-12 is proved by substitution, property, segment addition postulate.

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Most popular questions from this chapter

Each figure in Exercises 19-24 is a parallelogram with its diagonals drawn. Find the values of x and y.

M, N and T are the midpoints of the sides of ΔXYZ.

State the theorem suggested by Exercise 6.

Given: parallelogram ABCD; DE¯⊥AC¯; BF¯⊥AC¯.

Prove: DEBF is a parallelogram.

Study the markings on each figure and decide whether ABCD must be a parallelogram. If the answer is yes, state the definition or theorem that applies.

Draw two segments that are both parallel and congruent. Connect their endpoints to form a quadrilateral. What appears to be true of the quadrilateral?
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