Question

The perpendicular bisector of side
AB
of ∆ABC intersects the extension of side
AC
at D. Find the measure of ∠ABC if m∠CBD = 16° and m∠ACB = 118°.

Answer 1

Answer:

$m\angle ABC =23$°

Step-by-step explanation:

Let the measure of $m\angle ABC=x$

Let the perpendicular bisector of AB meet AB at E as shown in the figure below.

Given:

$m\angle CBD=16$°, $m\angle ACB=118$°

Consider the triangles Δ AED and Δ BED

AE ≅ BE  (Perpendicular bisector bisects the side equally)

$m\angle AED\cong m\angle BED= 90$° (Perpendicular bisector bisects the side at right angles)

ED ≅ ED ( Reflexive property)

Therefore, Δ AED ≅ Δ BED by SAS postulate.

Now, by CPCTE,

$m\angle BAC=m\angle ABD\\m\angle BAC=m\angle ABC+m\angle CBD\\m\angle BAC=x+16$

Now, consider the triangle Δ ABC,

The sum of all its interior angles is equal to 180°. Therefore,

$m\angle BAC + m\angle ABC+m\angle ACB=180\\x+16+x+118=180\\2x+134=180\\2x=180-134\\2x=46\\x=\frac{46}{2}=23$

Therefore, the measure of $m\angle ABC = x = 23$°.

Answer 2

m \ angle ABC = 23 °

Further Explanation

Let the size of m \ angle ABC = x

Let the perpendicular bisector of AB meet AB on E as shown in the figure below.

Granted:

m \ CBD angle = 16 °, m \ ACB angle = 118 °

Consider the triangles Δ AED and Δ BED

AE ≅ BE (perpendicular bisector bisects the sides equally)

m \ AED angle \ cong m \ BED angle = 90 ° (perpendicular bisector bisects the sides at right angles)

ED ≅ ED (reflexive property)

Therefore, Δ AED ≅ ED BED by SAS postulates.

Now, by CPCTE,

m \ angle BAC = m \ angle ABD

m \ angle BAC = m \ angle ABC m \ angle CBD

m \ angle BAC = x 16

Now, consider the triangle Δ ABC,

The sum of all interior angles is equal to 180 °. Therefore,

m \ BAC angle m \ ABC angle m \ angle ACB = 180

x 16 x 118 = 180

2x 134 = 180

2x = 180-134

2x = 46

x = \ frac {46} {2} = 23

Therefore, the size of m \ angle ABC = x = 23 °.

Learn More

Angles  https://brainly.com/question/13890076

Triangle  https://brainly.com/question/13890076

Details

Grade: Middle School

Subject: Mathematics

Keyword: angles, triangle, perpendicular