Chapter 6: Q12. (page 207)

Write proof in two-column form.
Given: $\bar{Q R}$ , $\bar{S T}$ bisect each other. Prove: $∠ X R T > m ∠ S$ .

Short Answer

Expert verified

Statement	Proof
1. $Q V = V R$ , $S V = V T$	Given
2. $∠ Q V S = ∠ R V T$	Vertically opposite angles
3. $△ Q S V ≅ △ V R T$	SAS
4. $∠ T = ∠ S$	CPCT
5. $∠ X R T = ∠ R V T + ∠ S$
6. $m ∠ X R T > m ∠ S$	Property of Inequality If $a = b + c$ and $c > 0$ then $a > c$ .

Step by step solution

Step 1. Draw the given diagram and define the congruent property of triangles

Two triangles $Q S V$ and $V R T$ as shown below are the given triangles.

If two sides and one angle equal of two triangles are equal then from SAS corresponding triangles will be congruent triangles.

Since QR and ST bisect each other, therefore $Q V = V R$ , $S V = V T$ .

The vertically opposite angles are always equal.

Therefore, $∠ Q V S = ∠ R V T$ . Then by SAS, $△ Q S V ≅ △ V R T$ .

Step 2. Property of congruent triangles

Since $△ Q S V ≅ △ V R T$ then by CPCT $∠ T = ∠ S$ .

Use exterior angle property as follows:

$\begin{array}{l} ∠ X R T = ∠ R V T + ∠ T \\ ∠ X R T = ∠ R V T + ∠ S \end{array}$

Step 3. Prove the statement

If $a = b + c$ and $c > 0$ then $a > c$ .

Since $∠ X R T = ∠ R V T + ∠ S$ therefore $m ∠ X R T > m ∠ S$

Hence proved.

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Write proof in two-column form.
Given: $\bar{Q R}$ , $\bar{S T}$ bisect each other. Prove: $∠ X R T > m ∠ S$ .

Short Answer

Step by step solution

Step 1. Draw the given diagram and define the congruent property of triangles

Step 2. Property of congruent triangles

Step 3. Prove the statement

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Write proof in two-column form.Given: QR¯, ST¯bisect each other. Prove: ∠XRT>m∠S.

Short Answer

Step by step solution

Step 1. Draw the given diagram and define the congruent property of triangles

Step 2. Property of congruent triangles

Step 3. Prove the statement

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Calculus

Discrete Mathematics

Geometry

Applied Mathematics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Write proof in two-column form.
Given: $\bar{Q R}$ , $\bar{S T}$ bisect each other. Prove: $∠ X R T > m ∠ S$ .