directions: find the missing measures in each figure. keep the angle relationships in mind.
1. diagram with 112° and y°
2. diagram with right angle, 68°, x°
3. diagram with 124° and x°
4. diagram with 43°, x°, y°, z°
5. diagram with right angle, 72°, x°, y°, z°
6. ∠1 and ∠2 are vertical angles. if the measure of ∠2 is 105°, find the measure of ∠1.
7. ∠a and ∠b are complementary angles. if the measure of ∠a is 42°, find the measure of ∠b.
8. ∠p and ∠q are supplementary angles. if the measure of ∠q is 64°, find the measure of ∠p.
9. ∠1 and ∠2 form a linear pair. if the measure of ∠1 is 113°, find the measure of ∠2.
10. using algebra to solve for missing measures
if ( mangle pqt = 3x + 47 ) and ( mangle sqr = 6x - 25 ), find the measure of ( angle sqr ).
diagram with intersecting lines at ( q ) (points ( p, t, s, r ))
11. if ( overleftrightarrow{ab} perp overleftrightarrow{cd} ), ( mangle dce = 7x + 2 ) and ( mangle ecb = x + 8 ), find the measure of ( angle dce ).
diagram with ( a ext{-}c ext{-}b ) horizontal, ( d ) vertical up from ( c ), ( e ) between ( d ) and ( b )
12. if ( mangle knm = 8x - 5 ) and ( mangle mnj = 4x - 19 ), find the measure of ( angle knm ).
diagram with intersecting lines at ( n ) (points ( l, k, j, m ))

Question

directions: find the missing measures in each figure. keep the angle relationships in mind.
1. diagram with 112° and y°
2. diagram with right angle, 68°, x°
3. diagram with 124° and x°
4. diagram with 43°, x°, y°, z°
5. diagram with right angle, 72°, x°, y°, z°
6. ∠1 and ∠2 are vertical angles. if the measure of ∠2 is 105°, find the measure of ∠1.
7. ∠a and ∠b are complementary angles. if the measure of ∠a is 42°, find the measure of ∠b.
8. ∠p and ∠q are supplementary angles. if the measure of ∠q is 64°, find the measure of ∠p.
9. ∠1 and ∠2 form a linear pair. if the measure of ∠1 is 113°, find the measure of ∠2.
10. using algebra to solve for missing measures
    if ( mangle pqt = 3x + 47 ) and ( mangle sqr = 6x - 25 ), find the measure of ( angle sqr ).
    diagram with intersecting lines at ( q ) (points ( p, t, s, r ))
11. if ( overleftrightarrow{ab} perp overleftrightarrow{cd} ), ( mangle dce = 7x + 2 ) and ( mangle ecb = x + 8 ), find the measure of ( angle dce ).
    diagram with ( a	ext{-}c	ext{-}b ) horizontal, ( d ) vertical up from ( c ), ( e ) between ( d ) and ( b )
12. if ( mangle knm = 8x - 5 ) and ( mangle mnj = 4x - 19 ), find the measure of ( angle knm ).
    diagram with intersecting lines at ( n ) (points ( l, k, j, m ))

Accepted Answer

### Problem 1
# Explanation:
## Step1: Identify vertical angles
Vertical angles are equal. The given angle is $112^\circ$, so $y = 112$ (vertical angles are congruent).
# Answer:
$y = 112$

### Problem 2
# Explanation:
## Step1: Identify complementary angles
The two angles form a right angle ($90^\circ$), so they are complementary. Let the unknown angle be $x$. Then $x + 68 = 90$.
## Step2: Solve for $x$
Subtract 68 from both sides: $x = 90 - 68 = 22$.
# Answer:
$x = 22$

### Problem 3
# Explanation:
## Step1: Identify supplementary angles
The two angles form a linear pair (supplementary), so their sum is $180^\circ$. Let the unknown angle be $x$. Then $124 + x = 180$.
## Step2: Solve for $x$
Subtract 124 from both sides: $x = 180 - 124 = 56$.
# Answer:
$x = 56$

### Problem 4
# Explanation:
## Step1: Identify vertical and supplementary angles
- $z$ and $43^\circ$ are supplementary (linear pair), so $z + 43 = 180$ ⇒ $z = 180 - 43 = 137$.
- $x$ and $43^\circ$ are vertical angles? No, $x$ and $z$ are supplementary? Wait, $x$ and $43^\circ$: actually, $x$ and $43^\circ$ are vertical? Wait, no, the angle $x$ and $43^\circ$: looking at the diagram, $x$ and $43^\circ$ are vertical angles? Wait, no, the horizontal line and the transversal: $x$ and $43^\circ$ are vertical? Wait, maybe $x = 43$ (vertical angles), $y = z = 137$ (since $y$ and $z$ are vertical, and $z$ is supplementary to $43$). Wait, let's re - check:
  - $z$ and $43^\circ$ form a linear pair, so $z = 180 - 43 = 137^\circ$.
  - $x$ and $43^\circ$ are vertical angles, so $x = 43^\circ$.
  - $y$ and $z$ are vertical angles, so $y = z = 137^\circ$.
# Answer:
$x = 43$, $y = 137$, $z = 137$

### Problem 5
# Explanation:
## Step1: Identify right angle and angle relationships
There is a right angle ($90^\circ$), and an angle of $72^\circ$.
- $z$ and $72^\circ$ are complementary (since they form a right angle), so $z + 72 = 90$ ⇒ $z = 90 - 72 = 18$.
- $x$ and $72^\circ$ are supplementary (linear pair), so $x + 72 = 180$ ⇒ $x = 180 - 72 = 108$.
- $y$ is a right angle? Wait, $y$ and the right angle: $y = 90^\circ$ (vertical to the right angle).
# Answer:
$x = 108$, $y = 90$, $z = 18$

### Problem 6
# Explanation:
## Step1: Recall vertical angles property
Vertical angles are congruent. So if $\angle2 = 105^\circ$, then $\angle1=\angle2 = 105^\circ$.
# Answer:
$m\angle1 = 105^\circ$

### Problem 7
# Explanation:
## Step1: Recall complementary angles property
Complementary angles sum to $90^\circ$. So $m\angle A+m\angle B = 90^\circ$. Given $m\angle A = 42^\circ$, then $m\angle B=90 - 42 = 48^\circ$.
# Answer:
$m\angle B = 48^\circ$

### Problem 8
# Explanation:
## Step1: Recall supplementary angles property
Supplementary angles sum to $180^\circ$. So $m\angle P+m\angle Q = 180^\circ$. Given $m\angle Q = 64^\circ$, then $m\angle P = 180 - 64 = 116^\circ$.
# Answer:
$m\angle P = 116^\circ$

### Problem 9
# Explanation:
## Step1: Recall linear pair property
A linear pair of angles is supplementary (sum to $180^\circ$). So $m\angle1+m\angle2 = 180^\circ$. Given $m\angle1 = 113^\circ$, then $m\angle2=180 - 113 = 67^\circ$.
# Answer:
$m\angle2 = 67^\circ$

### Problem 10
# Explanation:
## Step1: Identify vertical angles
$\angle PQT$ and $\angle SQR$ are vertical angles, so they are equal. Thus, $3x + 47=6x - 25$.
## Step2: Solve for $x$
Subtract $3x$ from both sides: $47 = 3x - 25$. Add 25 to both sides: $72 = 3x$. Divide by 3: $x = 24$.
## Step3: Find $m\angle SQR$
Substitute $x = 24$ into $6x - 25$: $6(24)-25=144 - 25 = 119$.
# Answer:
$m\angle SQR = 119^\circ$

### Problem 11
# Explanation:
## Step1: Identify right angle (perpendicular lines)
Since $\overrightarrow{AB}\perp\overrightarrow{CD}$, $\angle DCE+\angle ECB = 90^\circ$. So $(7x + 2)+(x + 8)=90$.
## Step2: Solve for $x$
Combine like terms: $8x+10 = 90$. Subtract 10: $8x = 80$. Divide by 8: $x = 10$.
## Step3: Find $m\angle DCE$
Substitute $x = 10$ into $7x + 2$: $7(10)+2=70 + 2 = 72$.
# Answer:
$m\angle DCE = 72^\circ$

### Problem 12
# Explanation:
## Step1: Identify supplementary angles (linear pair)
$\angle KNM$ and $\angle MNJ$ form a linear pair, so $m\angle KNM+m\angle MNJ = 180^\circ$. Thus, $(8x - 5)+(4x - 19)=180$.
## Step2: Solve for $x$
Combine like terms: $12x-24 = 180$. Add 24: $12x = 204$. Divide by 12: $x = 17$.
## Step3: Find $m\angle KNM$
Substitute $x = 17$ into $8x - 5$: $8(17)-5=136 - 5 = 131$.
# Answer:
$m\angle KNM = 131^\circ$

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 1

Step1: Identify vertical angles

Step1: Identify complementary angles

Step2: Solve for \(x\)

Step1: Identify supplementary angles

Step2: Solve for \(x\)

Answer:

Problem 2