Question

AD is a perpendicular bisector that goes through the

vertex of the isosceles triangle.

m

10. m<2 =
11. m<3 =
12. m<4 =
13. m<5 =
14. m<6 =

Answer 1

m∠2 = 90°

m∠3 = 35°

m∠4 = 35°

m∠5 = 90°

m∠6 = 55°

Explanation:

A perpendicular bisector is a line that intersects another line segment at 90°, dividing it into two equal parts.

If AD is a perpendicular bisector then:

⇒ m∠2 = m∠5 = 90°

An isosceles triangle has two legs of equal length.

Therefore, as AC = AB then AB is the angle bisector of ∠CAB.

Given m∠CAB = 70°:

⇒ m∠3 = m∠4 = 70° ÷ 2 = 35°

The interior angles of a triangle sum to 180°.

⇒ m∠4 + m∠5 + m∠6 = 180°

⇒ 35° + 90° + m∠6 = 180°

⇒ 125° + m∠6 = 180°

⇒ m∠6 = 180° - 125°

⇒ m∠6 = 55°

Answer 2

• m∠2 = 90°,
• m∠3 = 35°,
• m∠4 = 35°,
• m∠5 = 90°,
• m∠6 = 55°.

=========================

Given

• AD⊥BC,
• AD is bisector of ∠A,
• ΔABC is isosceles,
• ∠CAB = 70°.

Solution

According to the given above we conclude:

• AD⊥BC ⇒ 2 and 5 are right angles,
• AD is bisector of ∠A ⇒ 3 and 4 are equal angles, and add up to 70°.
• ΔABC is isosceles ⇒ 1 and 6 are equal angles

So the angles are:

• m∠2 = m∠5 = 90°,
• m∠3 = m∠4 = 35°,
• m∠6 = 90° - m∠4 = 90° - 35° = 55° (complementary angles)