example 1
1. find the measures of two supplementary angles if the difference between the measures of the two angles is 35°.
2. ∠e and ∠f are complementary. the measure of ∠e is 54° more than the measure of ∠f. find the measure of each angle.
3. the measure of an angles supplement is 76° less than the measure of the angle. find the measures of the angle and its supplement.
4. ∠q and ∠r are complementary. the measure of ∠q is 26° less than the measure of ∠r. find the measure of each angle.
5. the measure of the supplement of an angle is three times the measure of the angle. find the measures of the angle and its supplement.
6. the bascule bridge shown is opening from its horizontal position to its fully vertical position. so far, the bridge has lifted 35° in 21 seconds. at this rate, how much longer will it take for the bridge to reach its vertical position?
example 2
7. rays ba and bc are perpendicular. point d lies in the interior of ∠abc. if m∠abd=(3r + 5)° and m∠dbc=(5r - 27)°, find m∠abd and m∠dbc.
8. (overleftrightarrow{wx}) and (overleftrightarrow{yz}) intersect at point v. if m∠wvy=(4a + 58)° and m∠xvy=(2b - 18)°, find the values of a and b such that (overleftrightarrow{wx}) is perpendicular to (overleftrightarrow{yz}).
9. refer to the figure at the right. if m∠2=(a + 15)° and m∠3=(a + 35)°, find the value of a such that (overrightarrow{hl}perpoverrightarrow{hj}).
10. rays da and dc are perpendicular. point b lies in the interior of ∠adc. if m∠adb=(3a + 10)° and m∠bdc = 13a°, find a, m∠adb, and m∠bdc.

Question

Accepted Answer

1.
# Explanation:
## Step1: Let the two supplementary angles be $x$ and $y$ ($x>y$). Supplementary - angle property
We know that $x + y=180^{\circ}$ and $x - y = 35^{\circ}$.
## Step2: Add the two equations
$(x + y)+(x - y)=180^{\circ}+35^{\circ}$, which simplifies to $2x=215^{\circ}$, so $x = 107.5^{\circ}$.
## Step3: Find $y$
Substitute $x = 107.5^{\circ}$ into $x + y=180^{\circ}$, we get $y=180^{\circ}-107.5^{\circ}=72.5^{\circ}$.
# Answer:
The two angles are $107.5^{\circ}$ and $72.5^{\circ}$

2.
# Explanation:
## Step1: Let the measure of $\angle F=x$. Complementary - angle property
Since $\angle E$ and $\angle F$ are complementary, $\angle E+\angle F = 90^{\circ}$, and $\angle E=x + 54^{\circ}$.
## Step2: Substitute and solve
$(x + 54^{\circ})+x=90^{\circ}$, $2x=90^{\circ}-54^{\circ}=36^{\circ}$, so $x = 18^{\circ}$.
## Step3: Find $\angle E$
$\angle E=18^{\circ}+54^{\circ}=72^{\circ}$
# Answer:
$\angle E = 72^{\circ}$, $\angle F=18^{\circ}$

3.
# Explanation:
## Step1: Let the angle be $x$. Supplementary - angle property
Its supplement is $180^{\circ}-x$, and $180^{\circ}-x=x - 76^{\circ}$.
## Step2: Solve for $x$
Add $x$ to both sides: $180^{\circ}=2x - 76^{\circ}$, then $2x=180^{\circ}+76^{\circ}=256^{\circ}$, so $x = 128^{\circ}$.
## Step3: Find the supplement
The supplement is $180^{\circ}-128^{\circ}=52^{\circ}$
# Answer:
The angle is $128^{\circ}$ and its supplement is $52^{\circ}$

4.
# Explanation:
## Step1: Let the measure of $\angle R=x$. Complementary - angle property
Since $\angle Q$ and $\angle R$ are complementary, $\angle Q+\angle R = 90^{\circ}$, and $\angle Q=x - 26^{\circ}$.
## Step2: Substitute and solve
$(x - 26^{\circ})+x=90^{\circ}$, $2x=90^{\circ}+26^{\circ}=116^{\circ}$, so $x = 58^{\circ}$.
## Step3: Find $\angle Q$
$\angle Q=58^{\circ}-26^{\circ}=32^{\circ}$
# Answer:
$\angle Q = 32^{\circ}$, $\angle R=58^{\circ}$

5.
# Explanation:
## Step1: Let the angle be $x$. Supplementary - angle property
Its supplement is $180^{\circ}-x$, and $180^{\circ}-x = 3x$.
## Step2: Solve for $x$
Add $x$ to both sides: $180^{\circ}=4x$, so $x = 45^{\circ}$.
## Step3: Find the supplement
The supplement is $180^{\circ}-45^{\circ}=135^{\circ}$
# Answer:
The angle is $45^{\circ}$ and its supplement is $135^{\circ}$

6.
# Explanation:
## Step1: Find the remaining angle
The bridge needs to move from $35^{\circ}$ to $90^{\circ}$, so the remaining angle is $90^{\circ}-35^{\circ}=55^{\circ}$.
## Step2: Find the rate of movement
The rate of movement is $\frac{35^{\circ}}{21}=\frac{5}{3}$ degrees per second.
## Step3: Calculate the time
Time $t=\frac{55^{\circ}}{\frac{5}{3}\text{ degrees per second}}=55\times\frac{3}{5}=33$ seconds.
# Answer:
33 seconds

7.
# Explanation:
## Step1: Use the right - angle property
Since $\angle ABC = 90^{\circ}$ and $\angle ABC=\angle ABD+\angle DBC$, we have $(3r + 5)+(5r-27)=90$.
## Step2: Combine like terms and solve
$8r-22 = 90$, $8r=90 + 22=112$, so $r = 14$.
## Step3: Find the angles
$\angle ABD=3r + 5=3\times14 + 5=47^{\circ}$, $\angle DBC=5r-27=5\times14-27=43^{\circ}$
# Answer:
$\angle ABD = 47^{\circ}$, $\angle DBC=43^{\circ}$

8.
# Explanation:
## Step1: Use the perpendicular - angle property
If $\overleftrightarrow{WX}\perp\overleftrightarrow{YZ}$, then $\angle WVY = 90^{\circ}$ or $\angle XVY=90^{\circ}$.
For $\angle WVY = 4a + 58=90$, $4a=90 - 58 = 32$, $a = 8$.
For $\angle XVY=2b-18 = 90$, $2b=90 + 18=108$, $b = 54$
# Answer:
$a = 8$, $b = 54$

9.
# Explanation:
## Step1: Use the right - angle property
Since $\overrightarrow{HL}\perp\overrightarrow{HJ}$, $\angle 2+\angle 3=90^{\circ}$.
$(a + 15)+(a + 35)=90$.
## Step2: Combine like terms and solve
$2a+50 = 90$, $2a=90 - 50=40$, so $a = 20$
# Answer:
$a = 20$

10.
# Explanation:
## Step1: Use the right - angle property
Since $\angle ADC = 90^{\circ}$ and $\angle ADC=\angle ADB+\angle BDC$, we have $(3a + 10)+13a=90$.
## Step2: Combine like terms and solve
$16a+10 = 90$, $16a=90 - 10=80$, so $a = 5$.
## Step3: Find the angles
$\angle ADB=3a + 10=3\times5+10=25^{\circ}$, $\angle BDC=13a=13\times5 = 65^{\circ}$
# Answer:
$a = 5$, $\angle ADB = 25^{\circ}$, $\angle BDC=65^{\circ}$

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QUESTION IMAGE

Question

Explanation:

Step1: Let the two supplementary angles be \(x\) and \(y\) (\(x>y\)). Supplementary - angle property

Step2: Add the two equations

Step3: Find \(y\)

Step1: Let the measure of \(\angle F=x\). Complementary - angle property

Step2: Substitute and solve

Step3: Find \(\angle E\)

Step1: Let the angle be \(x\). Supplementary - angle property

Step2: Solve for \(x\)

Step3: Find the supplement

Answer: