if isosceles triangle abc has a 130° angle at vertex b, which statement must be true?\n○ m∠a = 15° and m∠c =…

if isosceles triangle abc has a 130° angle at vertex b, which statement must be true?\n○ m∠a = 15° and m∠c = 35°\n○ m∠a + m∠b = 155°\n○ m∠a + m∠c = 60°\n○ m∠a = 20° and m∠c = 30°

if isosceles triangle abc has a 130° angle at vertex b, which statement must be true?\n○ m∠a = 15° and m∠c = 35°\n○ m∠a + m∠b = 155°\n○ m∠a + m∠c = 60°\n○ m∠a = 20° and m∠c = 30°

Answer

Explanation:

Step1: Recall the properties of an isosceles triangle and the sum of angles in a triangle

In an isosceles triangle, two angles are equal. The sum of the interior angles of a triangle is (180^{\circ}). Given (\angle B = 130^{\circ}). Since (130^{\circ}) is an obtuse angle, it cannot be one of the equal angles (because if there were two angles of (130^{\circ}), (130 + 130=260>180)). So (\angle A=\angle C).

Step2: Calculate (\angle A) and (\angle C)

Let (\angle A = \angle C=x). Using the angle - sum formula for a triangle: (\angle A+\angle B+\angle C = 180^{\circ}). Substitute (\angle B = 130^{\circ}) and (\angle A=\angle C=x) into the formula: (x + 130+x=180). Combine like terms: (2x=180 - 130). So (2x = 50), and (x = 25^{\circ}).

Step3: Check each option

  • Option 1: (m\angle A = 15^{\circ}) and (m\angle C = 35^{\circ}). Since (\angle A=\angle C), this is false.
  • Option 2: (m\angle A+m\angle B). Since (m\angle A = 25^{\circ}) and (m\angle B=130^{\circ}), (m\angle A + m\angle B=25 + 130=155^{\circ}).
  • Option 3: (m\angle A+m\angle C). Since (m\angle A=m\angle C = 25^{\circ}), (m\angle A + m\angle C=25+25 = 50^{\circ}\neq60^{\circ}).
  • Option 4: (m\angle A = 20^{\circ}) and (m\angle C = 30^{\circ}). Since (\angle A=\angle C), this is false.

Answer:

(m\angle A + m\angle B=155^{\circ})