Triangles

70°

Step 1: Given triangle ABC is isosceles with AB = AC and angle A = 40°. Step 2: In an isosceles triangle, angles opposite to equal sides are equal. Step 3: Since AB = AC, angles B and C are equal (both opposite to equal sides). Step 4: Using angle sum property: angle A + angle B + angle C = 180°. Step 5: Substituting: 40° + angle B + angle B = 180°, so 2(angle B) = 140°, therefore angle B = 70°.

5 cm, 12 cm, 13 cm

Step 1: For any triangle, the sum of any two sides must be greater than the third side (triangle inequality). Step 2: Check option A: 3 + 4 = 7 < 8, violates triangle inequality. Step 3: Check option B: 5 + 12 = 17 > 13 ✓, 5 + 13 = 18 > 12 ✓, 12 + 13 = 25 > 5 ✓. Step 4: Check option C: 2 + 3 = 5 < 6, violates triangle inequality. Step 5: Check option D: 1 + 2 = 3 < 4, violates triangle inequality. Therefore, only option B satisfies triangle inequality.

8 cm

Step 1: Given right triangle ABC with right angle at C, AB = 10 cm (hypotenuse), BC = 6 cm. Step 2: Using Pythagoras theorem: (hypotenuse)² = (side₁)² + (side₂)². Step 3: AB² = AC² + BC², so 10² = AC² + 6². Step 4: 100 = AC² + 36, therefore AC² = 100 - 36 = 64. Step 5: AC = √64 = 8 cm.

ASA

Step 1: Given angle A = angle P, angle B = angle Q, and side AB = side PQ. Step 2: We have two pairs of equal angles and one pair of equal sides. Step 3: The equal side AB = PQ is between the two equal angles (angle A and angle B in triangle ABC, angle P and angle Q in triangle PQR). Step 4: This matches the ASA (Angle-Side-Angle) congruence rule. Step 5: Therefore, triangles ABC and PQR are congruent by ASA rule.