1. **Problem:** If $\triangle PQR \cong \triangle STV$, which sides and angles have equal measurements? 2. **Solution:** - By definition of congruent triangles, corresponding sides and angles are equal. - Corresponding sides: $PQ = ST$, $QR = TV$, $PR = SV$ - Corresponding angles: $\angle P = \angle S$, $\angle Q = \angle T$, $\angle R = \angle V$ --- 2. **Problem:** Identify the congruence theorem/postulate for each pair of triangles (a) through (d). - Since the figures are not explicitly given here, typical congruence postulates are: - SAS (Side-Angle-Side): Two sides and the included angle are equal. - SSS (Side-Side-Side): All three sides are equal. - AAS (Angle-Angle-Side): Two angles and a non-included side are equal. - Based on the markings described: - (a) SAS - (b) SSS - (c) AAS - (d) SAS --- 3. **Problem:** In an isosceles triangle, the angle at the base is $45^\circ$. Find the angle opposite to the base. 4. **Solution:** - In an isosceles triangle, the angles opposite the equal sides are equal. - Let the equal sides be the legs, and the base angles each $45^\circ$. - Sum of angles in triangle: $180^\circ$ - Let the vertex angle be $x$. - Equation: $x + 45 + 45 = 180$ - Simplify: $x + 90 = 180$ - Solve: $x = 180 - 90 = 90$ - So, the angle opposite the base is $90^\circ$. --- 4. **Problem:** If the arms of two angles are parallel and both arms of each pair are in the same or opposite direction, prove these angles are congruent. 5. **Solution:** - When two angles have their arms parallel and oriented similarly or oppositely, they are either corresponding angles or alternate interior angles. - By the properties of parallel lines cut by a transversal, these angles are congruent. --- 5. **Problem:** Find the measure of sides of triangle where $m\angle B = m\angle C$ and sides are given as: - $A = (4x + 2)$ cm - $B = (6x - 8)$ cm - $C = (2x + 2)$ cm 6. **Solution:** - Since $m\angle B = m\angle C$, sides opposite these angles are equal. - Side opposite $\angle B$ is $AC$, side opposite $\angle C$ is $AB$. - Given sides $B$ and $C$ correspond to these sides, so set $B = C$: $$6x - 8 = 2x + 2$$ - Subtract $2x$ from both sides: $$6x - \cancel{8} - 2x = 2x + 2 - 2x$$ $$4x - 8 = 2$$ - Add 8 to both sides: $$4x = 10$$ - Divide both sides by 4: $$x = \frac{10}{4} = 2.5$$ - Find side lengths: $$A = 4(2.5) + 2 = 10 + 2 = 12$$ $$B = 6(2.5) - 8 = 15 - 8 = 7$$ $$C = 2(2.5) + 2 = 5 + 2 = 7$$ - So, sides are $A = 12$ cm, $B = 7$ cm, $C = 7$ cm. --- **Final answers:** 1. Corresponding sides and angles equal as $PQ=ST$, $QR=TV$, $PR=SV$ and $\angle P=\angle S$, $\angle Q=\angle T$, $\angle R=\angle V$. 2. (a) SAS, (b) SSS, (c) AAS, (d) SAS. 3. Angle opposite base is $90^\circ$. 4. Angles with parallel arms in same/opposite directions are congruent by parallel line angle properties. 5. Side lengths: $A=12$ cm, $B=7$ cm, $C=7$ cm.