Question

Angle Addition Postulate and Angle Bisector (pleaseee help me and show work)

Answer 1

The measures of angles determined using the angle addition postulate and the angle bisector theorem are:

1. $m \angle JIH = 145^{\circ}$

2. $\\m \angle IJK = 132^{\circ}$

3. $m \angle IBA = 120^{\circ}$

4. $m \angle VUE= 20^{\circ}$

5. $x = 4$

6. $x = -6$

7. $m \angle UVY = 30^{\circ}$

8. $m \angle IHG = 129^{\circ}$

1. Find $m \angle JIH$

• Given:

$m \angle JIE = 60^{\circ}\\m \angle EIH = 85^{\circ}$

• Therefore:

$m \angle JIH = m \angle JIE + m\angle EIH$ (angle addition postulate)

Substitute

$m \angle JIH = 60 + 85\\m \angle JIH = 145^{\circ}$

2. Find $m \angle IJK$

• Given:

$m \angle IJB = 82^{\circ}\\m \angle BJK = 50^{\circ}$

• Therefore:

$m \angle IJK = m \angle IJB + m\angle BJK$ (angle addition postulate)

Substitute

$m \angle IJK= 82 + 50\\m \angle IJK = 132^{\circ}$

3. Find $m \angle IBA$

• Given:

$m \angle CBI = 46^{\circ}\\m \angle CBA = 166^{\circ}$

• Therefore:

$m \angle CBA = m \angle IBA + m\angle CBI$ (angle addition postulate)

Substitute

$166 = m \angle IBA + 46$

Subtract 46 from both sides

$166 - 46 = m \angle IBA \\120 = m \angle IBA\\m \angle IBA = 120^{\circ}$

4. Find  $m \angle VUE$

• Given:

$m \angle VUT = 160^{\circ}\\m \angle EUT = 140^{\circ}$

• Therefore:

$m \angle VUT= m \angle EUT+ m\angle VUE$ (angle addition postulate)

Substitute

$160 = 140 + m \angle VUE$

Subtract 140 from both sides

$160 - 140 = m \angle VUE\\20 = m \angle VUE\\m \angle VUE= 20^{\circ}$

5. Find x

• Given:

$m \angle PQR = 86^{\circ}\\m \angle BQR = 11x + 6\\m \angle PQB = 10x + 4$

• Thus:

$m \angle BQR + m \angle PQB = m \angle PQR$ (Angle addition postulate)

Substitute

$11x + 6 + 10x - 4 = 86$

Add like terms and solve for x

$21x + 2 = 86\\21x = 86 - 2\\21x = 84\\x = 4$

6.  Find x

• Given:

$m \angle LMN= 160^{\circ}\\m \angle RMN= x + 46\\m \angle LMR= x + 126$

• Thus:

$m \angle LMR+ m \angle RMN= m \angle LMN$ (Angle addition postulate)

Substitute

$x + 126 + x + 46 = 160$

Add like terms and solve for x

$2x + 172 = 160$

$2x = 160 - 172\\2x = - 12\\x = -6$

7. Find $m \angle UVY$

• Given:

$m \angle YVW = 14x + 5\\m \angle UVY = 10 + 2x\\m \angle UVW = 175$

• First, create an equation to find the value of x.

$14x + 5 + 10 + 2x = 175$ (angle addition postulate)

Add like terms and solve for x

$16x + 15 = 175\\16x = 175 - 15\\16x = 160\\x = 10$

• Thus:

$m \angle UVY = 10 + 2x$

Plug in the value of x

$m \angle UVY = 10 + 2(10) \\m \angle UVY = 30^{\circ}$

8. Find $m \angle IHG$

• Given:

$m \angle IHG = 3 + 21x\\m \angle KHG = 64\\m \angle IHK = 10x + 5$

• First, create an equation to find the value of x.

$64 + 10x + 5 = 3 + 21x$ (angle addition postulate)

Add like terms and solve for x

$64 + 10x + 5 = 3 + 21x\\69 + 10x = 3 + 21x\\69 - 3 = -10x +21x\\66 = 11x\\6 = x\\x = 6$

• Thus:

$m \angle IHG = 3 + 21x$

Plug in the value of x

$m \angle IHG = 3 + 21(6)\\m \angle IHG = 129^{\circ}$

The answers are:

1. $m \angle JIH = 145^{\circ}$

2. $\\m \angle IJK = 132^{\circ}$

3. $m \angle IBA = 120^{\circ}$

4. $m \angle VUE= 20^{\circ}$

5. $x = 4$

6. $x = -6$

7. $m \angle UVY = 30^{\circ}$

8. $m \angle IHG = 129^{\circ}$

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