Question

\\(\delta abc \cong \delta rqp\\). using cpctc, find the congruent corresponding parts and their measures.

\\(pr = ca = \text{select}\\)
\\(rq = \text{select} = \text{select}\\)
\\(pq = \text{select} = \text{select}\\)

\\(m\angle p = m\angle \text{select} = \text{select}^\circ\\)
\\(m\angle r = m\angle \text{select} = \text{select}^\circ\\)
\\(m\angle q = m\angle \text{select} = \text{select}^\circ\\)

Explanation:

🆕 New Concept Discovered: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)
Matching corresponding vertices in congruent triangles to find equal sides and angles.

Step 1: Identify corresponding parts from the congruence statement

The problem states that:
\[ \Delta ABC \cong \Delta RQP \]

This tells us the exact correspondence between the vertices of the two triangles:

• Vertex \( A \) corresponds to vertex \( R \)
• Vertex \( B \) corresponds to vertex \( Q \)
• Vertex \( C \) corresponds to vertex \( P \)

Using this correspondence, we can pair up the congruent sides:

• \( AB \cong RQ \)
• \( BC \cong QP \) (or \( PQ \))
• \( AC \cong RP \) (or \( PR \))

And the congruent angles:

• \( \angle A \cong \angle R \)
• \( \angle B \cong \angle Q \)
• \( \angle C \cong \angle P \)

Step 2: Find the lengths of the sides

From the given image, we have the measurements for \( \Delta ABC \):

• \( AC = 9 \)
• \( BC = 6 \)

Using the corresponding side relationships:

• \( PR = CA = 9 \)
• \( RQ = AB \) (not given directly, but we can match the names)
• \( PQ = BC = 6 \)

Step 3: Find the measures of the angles

From the given image, we have the angle measures for \( \Delta ABC \):

• \( m\angle B = 75^\circ \)
• \( m\angle C = 59^\circ \)

We can find the third angle, \( m\angle A \), using the triangle angle sum theorem:
\[ m\angle A = 180^\circ - (75^\circ + 59^\circ) = 180^\circ - 134^\circ = 46^\circ \]

Now, match the corresponding angles to find their measures:

• \( m\angle P = m\angle C = 59^\circ \)
• \( m\angle R = m\angle A = 46^\circ \)
• \( m\angle Q = m\angle B = 75^\circ \)

Answer:

• \( PR = CA = 9 \)
• \( RQ = AB \)
• \( PQ = BC = 6 \)
• \( m\angle P = m\angle C = 59^\circ \)
• \( m\angle R = m\angle A = 46^\circ \)
• \( m\angle Q = m\angle B = 75^\circ \)