Question

Use the properties of triangle to find the unknown angles.
step by step pl​

Answer 1

Step-by-step explanation:

Solutions :-

Figure -1:-

In ∆ ABC , angle ABC = 40°

angle CAB = x°

and angle ACD = 50°

angle ACD is an exterior angle formed by extending BC to D

We know that

The exterior angle of a triangle formed by extending one side is equal to the sum of the opposite interior angles.

angle ACD = angle CAB + angle ABC

=> 50° = x° + 40°

=> x° = 50°-40°

=> x° = 10°

and

In ∆ ACD , AC = CD

=> angle CDA = angle CAD

Since the angles opposite to equal sides are equal.

Let angle CDA = angle CAD = A°

We know that

The sum of all angles in a triangle is 180°

In ∆ ACD,

angle CDA +angle CAD + angle ACD = 180°

=> A°+A°+50° = 180°

=> 2A°+50° = 180°

=> 2A° = 180°-50°

=> 2A° = 130°

=> A° = 130°/2

=> A° = 65°

now,

angle CDA = angle CAD = 65°

angle BAC + angle CAD+y = 180°

Since angles in the same line

=> 10°+65°+y = 180°

=> 75°+y =180°

=> y = 180°-75°

=> y = 105°

Therefore, x = 10° and y = 105°

Figure -2:-

In ∆ ABC, angle CAB = 65°

AB = AC => angle ACB = angle CBA

Since the angles opposite to equal sides are equal.

Let angle ACB = angle CBA = A°

We know that

The sum of all angles in a triangle is 180°

=> angle ACB + angle CBA + angle CAB = 180°

=> A°+A°+65° = 180°

=> 2A°+65° = 180°

=> 2A° = 180°-65°

=> 2A° = 115°

=> A° = 115°/2

=> A° = 57.5°

angle ACB = angle CBA = 57.5°

and

In ∆ BCD , angle BCD = 110°

and

BC = CD => angle BDC = angle CBD

Since the angles opposite to equal sides are equal.

Let angle BDC = angle CBD = M°

The sum of all angles in a triangle is 180°

angle BCD+angle BDC + angle CBD = 180°

=> M°+M°+110° = 180°

=> 2M°+110°° = 180°

=> 2M° = 180°-110°

=> 2M° = 70°

=> M° = 70°/2

=> M° = 35°

Therefore, angle BDC = angle CBD = 35°

Now,

angle B = angle ABC+ angle CBD

=> x = 57.5°+35°

=> x = 92.5°

therefore, x = 92.5°

Answer:-

Figure -1:-

x = 10° and y = 105°

Figure -2:-

x = 92.5°

Used formulae:-

• The sum of all angles in a triangle is 180°
• the angles opposite to equal sides are equal.
• The exterior angle of a triangle formed by extending one side is equal to the sum of the opposite interior angles.