Question

1. the two triangles are congruent. find all missing sides and angles.
2. if ( \triangle hij cong \triangle prq ), complete the congruency statements.

( overline{hi} cong )
( angle p cong )
( overline{pq} cong )
( angle jih cong )
( overline{jh} cong )
( angle j cong )

1. the two triangles are congruent. solve for ( x ). then find the missing sides.
2. the two triangles are congruent. solve for ( x ). then find the missing sides.

Explanation:

1.

Step1: Find the third angle of the first triangle

The sum of angles in a triangle is \(180^{\circ}\). Let the third angle be \(A\).
\(A = 180-(73 + 27)\)
\(A=180 - 100\)
\(A = 80^{\circ}\)
Since the triangles are congruent, the sides and angles correspond.
The side of length \(8\) in the first triangle corresponds to a side in the second triangle. The side of length \(11\) and \(13\) in the second triangle correspond to sides in the first triangle.
The angles \(73^{\circ}\), \(27^{\circ}\), \(80^{\circ}\) in the first triangle correspond to angles in the second triangle.

2.

When \(\triangle HIJ\cong\triangle PRQ\), by the property of congruent triangles (corresponding parts of congruent triangles are congruent - CPCTC):

• For sides: \(\overline{HI}\cong\overline{PR}\), \(\overline{PQ}\cong\overline{HJ}\), \(\overline{JH}\cong\overline{QR}\)
• For angles: \(\angle P\cong\angle H\), \(\angle JIH\cong\angle RQP\), \(\angle J\cong\angle Q\)

3.

Step1: Set up the equation for congruent sides

Since the two triangles are congruent, \(3x - 10=x + 32\)

Step2: Solve for \(x\)

Subtract \(x\) from both sides: \(3x-x-10=x - x+ 32\)
\(2x-10 = 32\)
Add \(10\) to both sides: \(2x-10 + 10=32 + 10\)
\(2x=42\)
Divide both sides by \(2\): \(x = 21\)

Step3: Find the missing sides

Substitute \(x = 21\) into \(3x-10\) (or \(x + 32\))
\(3(21)-10=63 - 10=53\)

4.

Answer:

1. The missing side of the first triangle is \(11\) (corresponding to the side of length \(11\) in the second triangle), the missing sides of the second triangle are \(8\) (corresponding to the side of length \(8\) in the first triangle). The angles of the second triangle are \(73^{\circ}\), \(27^{\circ}\), \(80^{\circ}\)
2. \(\overline{HI}\cong\overline{PR}\), \(\overline{PQ}\cong\overline{HJ}\), \(\overline{JH}\cong\overline{QR}\), \(\angle P\cong\angle H\), \(\angle JIH\cong\angle RQP\), \(\angle J\cong\angle Q\)
3. \(x = 21\), the missing sides are \(53\)
4. \(x = 9\), the missing sides are \(27\)