Math / Geometry

QuestionLesson 5.3 Example 3 Treat each red box of choices as a set of dropdown choices for each blank. Choose the best answer for each blank. You are given that $\overline{P Q} \cong \overline{P S}$ . By the Reflexive Property of Congruence Transitive Property of Congruence Symmetric Property of Congruence $\overline{R P} \cong \overline{R P}$ . definition of perpendicular lines congruent by the $\square$ triangle sum theorem the corresponding hypotenuse and leg of a right triangle three pairs of sides $\qquad$ So, $\qquad$ Two pairs of sides and their included angle . are congruent.
So, $\triangle P Q R$ and $\triangle P S R$ are congruent by the HL Congruence Theorem SAS Congruence Theorem SAS Congruence Theorem

Studdy Solution

STEP 1

1. We are given that $\overline{P Q} \cong \overline{P S}$ .
2. We need to determine the correct properties and theorems to prove the congruence of triangles.
3. The problem involves identifying the correct choices from dropdown options.

STEP 2

1. Identify the correct property of congruence for $\overline{R P} \cong \overline{R P}$ .
2. Determine the correct description of the congruence condition.
3. Choose the appropriate congruence theorem for $\triangle P Q R$ and $\triangle P S R$ .

STEP 3

Identify the correct property of congruence for $\overline{R P} \cong \overline{R P}$ :
The correct choice is the Reflexive Property of Congruence, which states that any geometric figure is congruent to itself.

STEP 4

Determine the correct description of the congruence condition:
Given that $\overline{P Q} \cong \overline{P S}$ and $\overline{R P} \cong \overline{R P}$ , we have two pairs of sides and their included angle congruent.

STEP 5

Choose the appropriate congruence theorem for $\triangle P Q R$ and $\triangle P S R$ :
The correct theorem is the SAS Congruence Theorem, which states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.
The solution is:
1. Reflexive Property of Congruence
2. Two pairs of sides and their included angle
3. SAS Congruence Theorem

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