Answer:
Part 9) ![x=7](https://tex.z-dn.net/?f=x%3D7)
Part 10) ![m\angle Y=92^o](https://tex.z-dn.net/?f=m%5Cangle%20Y%3D92%5Eo)
Part 11) ![x=5](https://tex.z-dn.net/?f=x%3D5)
Part 12)
![m\angle 1=60^o](https://tex.z-dn.net/?f=m%5Cangle%201%3D60%5Eo)
![m\angle 2=41^o](https://tex.z-dn.net/?f=m%5Cangle%202%3D41%5Eo)
![m\angle 3=19^o](https://tex.z-dn.net/?f=m%5Cangle%203%3D19%5Eo)
![m\angle 4=41^o](https://tex.z-dn.net/?f=m%5Cangle%204%3D41%5Eo)
![m\angle 5=19^o](https://tex.z-dn.net/?f=m%5Cangle%205%3D19%5Eo)
Part 13) The measure of each side is 56 units
Part 14)
![x=10/3](https://tex.z-dn.net/?f=x%3D10%2F3)
![HI=12\ units](https://tex.z-dn.net/?f=HI%3D12%5C%20units)
![HJ=12\ units](https://tex.z-dn.net/?f=HJ%3D12%5C%20units)
![IJ=38/3\ units](https://tex.z-dn.net/?f=IJ%3D38%2F3%5C%20units)
Part 15)
![x=7](https://tex.z-dn.net/?f=x%3D7)
![m\angle C=41^o](https://tex.z-dn.net/?f=m%5Cangle%20C%3D41%5Eo)
![m\angle D=41^o](https://tex.z-dn.net/?f=m%5Cangle%20D%3D41%5Eo)
![m\angle A=98^o](https://tex.z-dn.net/?f=m%5Cangle%20A%3D98%5Eo)
Step-by-step explanation:
Part 9) we know that
An isosceles triangle has two equal sides and two equal interior angles
The triangle of the problem is an isosceles triangle
Because has two equal sides and two equal angles
Remember
The sum of the interior angles in any triangle must be equal to 180 degrees
so
![2(8x-4)^o+76^o=180^o](https://tex.z-dn.net/?f=2%288x-4%29%5Eo%2B76%5Eo%3D180%5Eo)
solve for x
![16x-8+76=180\\16x=180-68\\16x=112\\x=7](https://tex.z-dn.net/?f=16x-8%2B76%3D180%5C%5C16x%3D180-68%5C%5C16x%3D112%5C%5Cx%3D7)
Part 10) we know that
step 1
Solve for x
An isosceles triangle has two equal sides and two equal interior angles
The triangle XYZ of the problem is an isosceles triangle
Because has two equal sides
so
![m\angle X=m\angle Z](https://tex.z-dn.net/?f=m%5Cangle%20X%3Dm%5Cangle%20Z)
substitute the given values
![(4x-12)^o=(2x+16)^o](https://tex.z-dn.net/?f=%284x-12%29%5Eo%3D%282x%2B16%29%5Eo)
solve for x
![4x-2x=16+12\\2x=28\\x=14](https://tex.z-dn.net/?f=4x-2x%3D16%2B12%5C%5C2x%3D28%5C%5Cx%3D14)
Find the measure of angle X
![m\angle X=(4x-12)^o](https://tex.z-dn.net/?f=m%5Cangle%20X%3D%284x-12%29%5Eo)
substitute the value of x
![m\angle X=(4(14)-12)=44^o](https://tex.z-dn.net/?f=m%5Cangle%20X%3D%284%2814%29-12%29%3D44%5Eo)
so
![m\angle Z=44^o](https://tex.z-dn.net/?f=m%5Cangle%20Z%3D44%5Eo)
step 2
Find the measure of angle Y
Remember
The sum of the interior angles in any triangle must be equal to 180 degrees
so
![m\angle Y+2(44^o)=180^o](https://tex.z-dn.net/?f=m%5Cangle%20Y%2B2%2844%5Eo%29%3D180%5Eo)
![m\angle Y=180^o-88^o=92^o](https://tex.z-dn.net/?f=m%5Cangle%20Y%3D180%5Eo-88%5Eo%3D92%5Eo)
Part 11) we know that
we know that
An equilateral triangle has three equal sides and three equal interior angles. The measure of each interior angle is 60 degrees
In this problem we have an equilateral triangle
so
![(14x-10)^o=60^o](https://tex.z-dn.net/?f=%2814x-10%29%5Eo%3D60%5Eo)
solve for x
![14x=60+10\\14x=70\\x=5](https://tex.z-dn.net/?f=14x%3D60%2B10%5C%5C14x%3D70%5C%5Cx%3D5)
Part 12)
step 1
Find the measure of angle 1
we know that
An equilateral triangle has three equal sides and three equal interior angles. The measure of each interior angle is 60 degrees
In this problem the triangle WXY is an equilateral triangle
therefore
![m\angle 1=60^o](https://tex.z-dn.net/?f=m%5Cangle%201%3D60%5Eo)
step 2
Find the measure of angle 3
we know that
Triangle WZY is an isosceles triangle
so
![m\angle 3=m\angle 5](https://tex.z-dn.net/?f=m%5Cangle%203%3Dm%5Cangle%205)
Remember that
The sum of the interior angles in any triangle must be equal to 180 degrees
so
![m\angle 3+m\angle 5+142^o=180^o](https://tex.z-dn.net/?f=m%5Cangle%203%2Bm%5Cangle%205%2B142%5Eo%3D180%5Eo)
![2m\angle 3=180^o-142^o](https://tex.z-dn.net/?f=2m%5Cangle%203%3D180%5Eo-142%5Eo)
![m\angle 3=19^o](https://tex.z-dn.net/?f=m%5Cangle%203%3D19%5Eo)
so
![m\angle 5=19^o](https://tex.z-dn.net/?f=m%5Cangle%205%3D19%5Eo)
step 3
Find the measure of angle 2
we know that
The measure of each interior angle of an equilateral triangle is equal to 60 degrees
so
![m\angle 2+m\angle 3=60^o](https://tex.z-dn.net/?f=m%5Cangle%202%2Bm%5Cangle%203%3D60%5Eo)
substitute the given value
![m\angle 2+19^o=60^o](https://tex.z-dn.net/?f=m%5Cangle%202%2B19%5Eo%3D60%5Eo)
![m\angle 2=60^o-19^o](https://tex.z-dn.net/?f=m%5Cangle%202%3D60%5Eo-19%5Eo)
![m\angle 2=41^o](https://tex.z-dn.net/?f=m%5Cangle%202%3D41%5Eo)
step 4
Find the measure of angle 4
we know that
The measure of each interior angle of an equilateral triangle is equal to 60 degrees
so
![m\angle 4+m\angle 5=60^o](https://tex.z-dn.net/?f=m%5Cangle%204%2Bm%5Cangle%205%3D60%5Eo)
substitute the given value
![m\angle 4+19^o=60^o](https://tex.z-dn.net/?f=m%5Cangle%204%2B19%5Eo%3D60%5Eo)
![m\angle 4=60^o-19^o](https://tex.z-dn.net/?f=m%5Cangle%204%3D60%5Eo-19%5Eo)
![m\angle 4=41^o](https://tex.z-dn.net/?f=m%5Cangle%204%3D41%5Eo)
Part 13) we know that
we know that
An equilateral triangle has three equal sides and three equal interior angles. The measure of each interior angle is 60 degrees
so
AB=BC=AC
solve the equation
AB=BC
substitute the given values
![5x-9=7x-35\\7x-5x=35-9\\2x=26\\x=13](https://tex.z-dn.net/?f=5x-9%3D7x-35%5C%5C7x-5x%3D35-9%5C%5C2x%3D26%5C%5Cx%3D13)
Find the measure of side AB
![AB=5x-9](https://tex.z-dn.net/?f=AB%3D5x-9)
substitute the value of x
![AB=5(13)-9=56\ units](https://tex.z-dn.net/?f=AB%3D5%2813%29-9%3D56%5C%20units)
Verify BC and AC
----> is correct
----> is correct
Part 14) we know that
An isosceles triangle has two equal sides and two equal angles
In this problem Triangle HIJ is an isosceles triangle
because
Angle I is equal to Angle J
so
HI=HJ
substitute the given values
![6x-8=3x+2](https://tex.z-dn.net/?f=6x-8%3D3x%2B2)
solve for x
![6x-3x=2+8\\3x=10\\x=10/3](https://tex.z-dn.net/?f=6x-3x%3D2%2B8%5C%5C3x%3D10%5C%5Cx%3D10%2F3)
Find the measure of each side
![HI=6(10/3) -8=12\ units](https://tex.z-dn.net/?f=HI%3D6%2810%2F3%29%20-8%3D12%5C%20units)
![HJ=3(10/3)+2=12\ units](https://tex.z-dn.net/?f=HJ%3D3%2810%2F3%29%2B2%3D12%5C%20units)
![IJ=5(10/3)-4=38/3\ units](https://tex.z-dn.net/?f=IJ%3D5%2810%2F3%29-4%3D38%2F3%5C%20units)
Part 15) we know that
Triangle ACD is an isosceles triangle
because
has two equal sides
AC=AD
so
![m\angle C=m\angle D](https://tex.z-dn.net/?f=m%5Cangle%20C%3Dm%5Cangle%20D)
substitute the given values
![6x-1=7x-8](https://tex.z-dn.net/?f=6x-1%3D7x-8)
solve for x
![7x-6x=8-1\\x=7](https://tex.z-dn.net/?f=7x-6x%3D8-1%5C%5Cx%3D7)
Find the measure of angle C
![m\angle C=6(7)-1=41^o](https://tex.z-dn.net/?f=m%5Cangle%20C%3D6%287%29-1%3D41%5Eo)
Find the measure of angle D
![m\angle D=7(7)-8=41^o](https://tex.z-dn.net/?f=m%5Cangle%20D%3D7%287%29-8%3D41%5Eo)
Find the measure of angle A
we have that
![m\angle C+m\angle D+m\angle A=180^o](https://tex.z-dn.net/?f=m%5Cangle%20C%2Bm%5Cangle%20D%2Bm%5Cangle%20A%3D180%5Eo)
substitute
![41^o+41^o+m\angle A=180^o](https://tex.z-dn.net/?f=41%5Eo%2B41%5Eo%2Bm%5Cangle%20A%3D180%5Eo)
![m\angle A=180^o-82^o](https://tex.z-dn.net/?f=m%5Cangle%20A%3D180%5Eo-82%5Eo)
![m\angle A=98^o](https://tex.z-dn.net/?f=m%5Cangle%20A%3D98%5Eo)