Question

for x, then find the measure of the angle given: 8. m∠abc__ m∠mpq 9. m∠mnp 10. m∠wxz__

Explanation:

Step1: Identify vertical - angle relationship

In the first pair of angles (3x + 23) and 4x, vertical angles are equal. So we set up the equation 3x+23 = 4x.
$3x + 23=4x$

Step2: Solve for x

Subtract 3x from both sides of the equation.
$4x-3x=23$
$x = 23$

Step3: Find m∠ABC

Since m∠ABC = 4x, substitute x = 23 into the expression.
$m\angle ABC=4\times23=92^{\circ}$

For the second problem:

Step1: Identify vertical - angle relationship

The angles 5x and 3x + 50 are vertical angles, so 5x=3x + 50.
$5x=3x + 50$

Step2: Solve for x

Subtract 3x from both sides:
$5x-3x=50$
$2x=50$
$x = 25$

Step3: Find m∠MPQ

Since m∠MPQ=3x + 50, substitute x = 25 into the expression.
$m\angle MPQ=3\times25+50=75 + 50=125^{\circ}$

For the third problem:

Step1: Identify vertical - angle relationship

The angles (a + 28) and 2a are vertical angles, so a + 28=2a.
$a + 28=2a$

Step2: Solve for a

Subtract a from both sides:
$2a-a=28$
$a = 28$

Step3: Find m∠MNP

Since m∠MNP=a + 28, substitute a = 28 into the expression.
$m\angle MNP=28+28=56^{\circ}$

For the fourth problem:
We assume the angles 5y and 2y+78 are vertical angles (since no other relationship is indicated), so 5y=2y + 78.

Step1: Solve for y

Subtract 2y from both sides:
$5y-2y=78$
$3y=78$
$y = 26$

Step2: Find m∠WXZ

Since m∠WXZ=5y, substitute y = 26 into the expression.
$m\angle WXZ=5\times26=130^{\circ}$

Answer:

m∠ABC = 92°
m∠MPQ = 125°
m∠MNP = 56°
m∠WXZ = 130°