13. if ( mangle deg = 5x - 4 ), ( mangle gef = 7x - 8 ), ( mangle deh = 9y + 5 ), find the values of ( x ) and ( y ).
14. ( angle r ) and ( angle s ) are complementary angles. if the ( mangle r = 12x - 3 ) and ( mangle s = 7x - 2 ), find ( mangle r ).
15. ( angle p ) and ( angle q ) are supplementary angles. if the ( mangle p = 4x + 1 ) and ( mangle q = 9x - 3 ), find ( mangle q ).
16. ( angle 1 ) and ( angle 2 ) form a linear pair. the measure of ( angle 2 ) is six more than twice the measure of ( angle 1 ). find ( mangle 2 ).
17. ( angle j ) and ( angle k ) are complementary angles. the measure of ( angle j ) is 18 less than the measure of ( angle k ). find the measures of both angles.
18. if ( overrightarrow{uw} ) bisects ( angle tuv ), ( mangle tuw = 13x - 5 ) and ( mangle wuv = 7x + 31 ), find the value of ( x ).
19. if ( overrightarrow{mo} ) bisects ( angle pmn ), ( mangle pmn = 74^circ ) and ( mangle omn = 2x + 7 ), find the value of ( x ).
20. if ( overrightarrow{ef} ) bisects ( angle ceb ), ( mangle cef = 7x + 21 ) and ( mangle feb = 10x - 3 ), find the measure of ( angle deb ).

Question

Accepted Answer

Let's solve each problem one by one:

### Problem 13:
From the diagram, $ \angle DEG $ and $ \angle GEF $ form a linear pair (they are adjacent and supplementary), and $ \angle DEH $ is vertical to $ \angle GEF $ (or supplementary to $ \angle DEG $, but also, since $ \angle DEH $ and $ \angle GEF $ are vertical angles? Wait, actually, looking at the diagram, $ \angle DEG + \angle GEF = 180^\circ $ (linear pair), and $ \angle DEH $ is equal to $ \angle GEF $ (vertical angles) or $ \angle DEG $? Wait, no, let's see:

First, $ \angle DEG $ and $ \angle GEF $ are adjacent and form a straight line, so:

$ m\angle DEG + m\angle GEF = 180^\circ $

Substitute the given expressions:

$ (5x - 4) + (7x - 8) = 180 $

Combine like terms:

$ 12x - 12 = 180 $

Add 12 to both sides:

$ 12x = 192 $

Divide by 12:

$ x = 16 $

Now, $ \angle DEH $ and $ \angle GEF $ are vertical angles? Wait, no, looking at the diagram, $ \angle DEH $ and $ \angle GEF $ are vertical angles? Wait, $ \angle DEG $ and $ \angle FEH $ are vertical, $ \angle GEF $ and $ \angle DEH $ are vertical? Wait, maybe $ \angle DEH $ is equal to $ \angle GEF $ because they are vertical angles. Wait, let's check:

If $ x = 16 $, then $ m\angle GEF = 7(16) - 8 = 112 - 8 = 104^\circ $

So $ m\angle DEH = 9y + 5 = 104 $ (since they are vertical angles)

Then:

$ 9y + 5 = 104 $

Subtract 5:

$ 9y = 99 $

Divide by 9:

$ y = 11 $

So $ x = 16 $, $ y = 11 $

### Problem 14:
$ \angle R $ and $ \angle S $ are complementary, so $ m\angle R + m\angle S = 90^\circ $

Substitute the expressions:

$ (12x - 3) + (7x - 2) = 90 $

Combine like terms:

$ 19x - 5 = 90 $

Add 5:

$ 19x = 95 $

Divide by 19:

$ x = 5 $

Now, find $ m\angle R $:

$ m\angle R = 12(5) - 3 = 60 - 3 = 57^\circ $

### Problem 15:
$ \angle P $ and $ \angle Q $ are supplementary, so $ m\angle P + m\angle Q = 180^\circ $

Substitute the expressions:

$ (4x + 1) + (9x - 3) = 180 $

Combine like terms:

$ 13x - 2 = 180 $

Add 2:

$ 13x = 182 $

Divide by 13:

$ x = 14 $

Now, find $ m\angle Q $:

$ m\angle Q = 9(14) - 3 = 126 - 3 = 123^\circ $

### Problem 16:
$ \angle 1 $ and $ \angle 2 $ form a linear pair, so $ m\angle 1 + m\angle 2 = 180^\circ $

Given that $ m\angle 2 = 2(m\angle 1) + 6 $

Let $ m\angle 1 = x $, then $ m\angle 2 = 2x + 6 $

So:

$ x + (2x + 6) = 180 $

Combine like terms:

$ 3x + 6 = 180 $

Subtract 6:

$ 3x = 174 $

Divide by 3:

$ x = 58 $

Then $ m\angle 2 = 2(58) + 6 = 116 + 6 = 122^\circ $

### Problem 17:
$ \angle J $ and $ \angle K $ are complementary, so $ m\angle J + m\angle K = 90^\circ $

Given that $ m\angle J = m\angle K - 18 $

Let $ m\angle K = x $, then $ m\angle J = x - 18 $

So:

$ (x - 18) + x = 90 $

Combine like terms:

$ 2x - 18 = 90 $

Add 18:

$ 2x = 108 $

Divide by 2:

$ x = 54 $

Thus, $ m\angle K = 54^\circ $, $ m\angle J = 54 - 18 = 36^\circ $

### Problem 18:
$ \overrightarrow{UW} $ bisects $ \angle TUV $, so $ m\angle TUW = m\angle WUV $

Substitute the expressions:

$ 13x - 5 = 7x + 31 $

Subtract $ 7x $ from both sides:

$ 6x - 5 = 31 $

Add 5:

$ 6x = 36 $

Divide by 6:

$ x = 6 $

### Problem 19:
$ \overrightarrow{MO} $ bisects $ \angle PMN $, so $ m\angle OMN = \frac{1}{2} m\angle PMN $

Given $ m\angle PMN = 74^\circ $, so $ m\angle OMN = \frac{74}{2} = 37^\circ $

But $ m\angle OMN = 2x + 7 $, so:

$ 2x + 7 = 37 $

Subtract 7:

$ 2x = 30 $

Divide by 2:

$ x = 15 $

### Problem 20:
$ \overrightarrow{EF} $ bisects $ \angle CEB $, so $ m\angle CEF = m\angle FEB $

Substitute the expressions:

$ 7x + 21 = 10x - 3 $

Subtract $ 7x $:

$ 21 = 3x - 3 $

Add 3:

$ 24 = 3x $

Divide by 3:

$ x = 8 $

Now, find $ m\angle CEF $ and $ m\angle FEB $:

$ m\angle CEF = 7(8) + 21 = 56 + 21 = 77^\circ $

$ m\angle FEB = 10(8) - 3 = 80 - 3 = 77^\circ $

Thus, $ m\angle CEB = 77 + 77 = 154^\circ $

Now, $ \angle DEB $ is vertical to $ \angle CEB $? Wait, looking at the diagram, $ \angle DEB $ and $ \angle CEA $ are vertical, but $ \angle DEB $ and $ \angle CEB $ are supplementary? Wait, no, $ \angle CEB $ and $ \angle DEB $ form a linear pair? Wait, no, the diagram shows that $ \angle CEB $ and $ \angle DEB $ are adjacent and form a straight line? Wait, no, actually, $ \angle CEB $ and $ \angle DEB $ are supplementary? Wait, no, let's check the diagram:

Points: A, E, B are on a line? Wait, the diagram has A, E, D and C, E, B? Wait, maybe $ \angle DEB $ is vertical to $ \angle CEA $, but $ \angle CEB $ and $ \angle DEB $ are supplementary? Wait, no, actually, since $ \angle CEB $ is 154°, then $ \angle DEB $ is 180° - 154° = 26°? Wait, no, maybe I made a mistake. Wait, $ \overrightarrow{EF} $ bisects $ \angle CEB $, so $ \angle CEB = 2 \times 77 = 154^\circ $. Then $ \angle DEB $ is vertical to $ \angle CEA $, but $ \angle CEA $ is vertical to $ \angle DEB $, and $ \angle CEB $ and $ \angle AED $ are vertical. Wait, maybe $ \angle DEB $ is supplementary to $ \angle CEB $? No, that doesn't make sense. Wait, maybe $ \angle DEB $ is equal to $ \angle CEA $, but $ \angle CEA $ is vertical to $ \angle DEB $. Wait, perhaps the diagram shows that $ \angle DEB $ is vertical to $ \angle CEA $, and $ \angle CEB $ and $ \angle AED $ are vertical. Wait, maybe I misread the diagram. Alternatively, maybe $ \angle DEB $ is supplementary to $ \angle CEB $, but that would be 180 - 154 = 26°, but let's check again.

Wait, the problem says "find the measure of $ \angle DEB $". Let's re-examine:

Since $ \overrightarrow{EF} $ bisects $ \angle CEB $, so $ m\angle CEF = m\angle FEB = 77^\circ $, so $ m\angle CEB = 154^\circ $. Now, $ \angle DEB $ and $ \angle CEB $ are adjacent and form a linear pair? Wait, no, the diagram has E as the intersection point, with lines AED and CEB intersecting at E. So $ \angle CEB $ and $ \angle AED $ are vertical angles, and $ \angle DEB $ and $ \angle CEA $ are vertical angles. Also, $ \angle CEB + \angle DEB = 180^\circ $? Wait, no, $ \angle CEB $ and $ \angle DEB $ are adjacent and form a straight line? Wait, if C, E, B are on a line, and D, E, B are on a line? No, the diagram shows A, E, D and C, E, B intersecting at E. So $ \angle CEB $ and $ \angle AED $ are vertical (154°), and $ \angle DEB $ and $ \angle CEA $ are vertical. Also, $ \angle CEB + \angle DEB = 180^\circ $? Wait, no, $ \angle CEB $ is 154°, so $ \angle DEB = 180 - 154 = 26^\circ $? Wait, that seems off. Wait, maybe $ \angle DEB $ is equal to $ \angle CEA $, and $ \angle CEA $ is 180 - 154 = 26°, so $ \angle DEB = 26^\circ $. Alternatively, maybe I made a mistake in the bisecting. Wait, no, the bisecting gives $ \angle CEF = \angle FEB = 77^\circ $, so $ \angle CEB = 154^\circ $. Then $ \angle DEB $ is supplementary to $ \angle CEB $, so 180 - 154 = 26°. So $ m\angle DEB = 26^\circ $.

### Final Answers:
13. $ x = 16 $, $ y = 11 $
14. $ m\angle R = 57^\circ $
15. $ m\angle Q = 123^\circ $
16. $ m\angle 2 = 122^\circ $
17. $ m\angle J = 36^\circ $, $ m\angle K = 54^\circ $
18. $ x = 6 $
19. $ x = 15 $
20. $ m\angle DEB = 26^\circ $

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 13:

Problem 14:

Problem 15:

Problem 16:

Problem 17:

Problem 18:

Problem 19:

Problem 20:

Answer:

Problem 13:

Problem 14:

Problem 15:

Problem 16:

Problem 17:

Problem 18:

Problem 19:

Problem 20:

Final Answers: