Chapter 3: Q32 (page 99)

Given: $P R$ bisects $∠ S P Q$ and $P S ⊥ S Q; R Q ⊥ P Q$ .
Which numbered angles must be congruent?

Short Answer

Expert verified

The congruent angles are $∠ 7 ≅ ∠ 8$ and $∠ 11 ≅ ∠ 12$ .

Step by step solution

Step 1. Apply the definition of angle bisector.

Angle bisector is a line which divides an angle into two equal parts.

From the figure, it can be observed that $\bar{P R}$ bisect $∠ S P Q$ . Therefore,

$∠ 7 ≅ ∠ 8$

Step 2. Description of step.

$\bar{P S} ⊥ \bar{S Q}$ and $\bar{R Q} ⊥ \bar{P Q}$ , therefore,

$\begin{matrix} ∠ Q ≅ ∠ S \\ m ∠ Q = 90 \end{matrix}$

Step 3. Description of step.

Consider $Δ P Q R and Δ P S T$ , in which $∠ Q ≅ ∠ S, ∠ R P Q ≅ ∠ S P T$ implies that $∠ P T S ≅ ∠ 11$ .

Step 4. Apply definition of vertically opposite angles.

Vertically opposite angles are the opposite angles formed due to intersection of two lines.

Consider $Δ P Q R and Δ P S T$ ,

$∠ 11 and ∠ 12$ are vertically opposite angles, therefore, $∠ 11 ≅ ∠ 12$ .

Therefore, the congruent angles are $∠ 7 ≅ ∠ 8$ and $∠ 11 ≅ ∠ 12$ .

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Given: $P R$ bisects $∠ S P Q$ and $P S ⊥ S Q; R Q ⊥ P Q$ .
Which numbered angles must be congruent?

Short Answer

Step by step solution

Step 1. Apply the definition of angle bisector.

Step 2. Description of step.

Step 3. Description of step.

Step 4. Apply definition of vertically opposite angles.

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Given: PRbisects ∠SPQand PS⊥SQ;RQ⊥PQ.Which numbered angles must be congruent?

Short Answer

Step by step solution

Step 1. Apply the definition of angle bisector.

Step 2. Description of step.

Step 3. Description of step.

Step 4. Apply definition of vertically opposite angles.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Decision Maths

Theoretical and Mathematical Physics

Applied Mathematics

Geometry

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Given: $P R$ bisects $∠ S P Q$ and $P S ⊥ S Q; R Q ⊥ P Q$ .
Which numbered angles must be congruent?