Question

1. among two supplementary angles, the measure of the larger angle is 44° more than measure of the smaller. find their measures.
2. can two angles be supplementary if both of them are:

(a) acute (b) obtuse (c) right

1. among two complementary angles, the measure of the smaller angle is 36° less than measure of the larger. find their measure.
2. identify the vertically opposite angles in the following figures:

(a) (b)

1. find the measures of all the marked angles in the following figures:

(a) (b) (c)

1. find the value of x in the following figures:

(a) (b) (c)

1. in the figure given alongside. op and oq are opposite rays:

(a) if x = 32°, what is the value of y?
(b) if y = 24°, what is the value of x?

1. determine all the marked angles in the following figure in which l || m.

Explanation:

Step1: Solve problem 8

Let the measure of the smaller supplementary - angle be $x$. Then the measure of the larger supplementary - angle is $x + 44^{\circ}$. Since two supplementary angles add up to $180^{\circ}$, we have the equation $x+(x + 44^{\circ})=180^{\circ}$.
Combining like terms: $2x+44^{\circ}=180^{\circ}$.
Subtract $44^{\circ}$ from both sides: $2x=180^{\circ}-44^{\circ}=136^{\circ}$.
Divide both sides by 2: $x = 68^{\circ}$.
The larger angle is $x + 44^{\circ}=68^{\circ}+44^{\circ}=112^{\circ}$.

Step2: Solve problem 9

(a) Acute angles are less than $90^{\circ}$. If both angles are acute, say $\alpha<90^{\circ}$ and $\beta<90^{\circ}$, then $\alpha+\beta<180^{\circ}$, so two acute angles cannot be supplementary.
(b) Obtuse angles are greater than $90^{\circ}$. If both angles are obtuse, say $\alpha>90^{\circ}$ and $\beta>90^{\circ}$, then $\alpha+\beta>180^{\circ}$, so two obtuse angles cannot be supplementary.
(c) Right - angles are equal to $90^{\circ}$. If $\alpha = 90^{\circ}$ and $\beta = 90^{\circ}$, then $\alpha+\beta=180^{\circ}$, so two right angles are supplementary.

Step3: Solve problem 10

Let the measure of the larger complementary angle be $x$. Then the measure of the smaller complementary angle is $x - 36^{\circ}$. Since two complementary angles add up to $90^{\circ}$, we have the equation $x+(x - 36^{\circ})=90^{\circ}$.
Combining like terms: $2x-36^{\circ}=90^{\circ}$.
Add $36^{\circ}$ to both sides: $2x=90^{\circ}+36^{\circ}=126^{\circ}$.
Divide both sides by 2: $x = 63^{\circ}$.
The smaller angle is $x - 36^{\circ}=63^{\circ}-36^{\circ}=27^{\circ}$.

Step4: Solve problem 11

(a) In the first figure, $\angle1$ and $\angle3$ are vertically - opposite angles, and $\angle2$ and $\angle4$ are vertically - opposite angles.
(b) In the second figure, $\angle a$ and $\angle d$, $\angle b$ and $\angle e$, $\angle c$ and $\angle f$ are vertically - opposite angles.

Step5: Solve problem 12

(a) If two angles are linear - pair, they are supplementary. If one angle is $85^{\circ}$, then the other angle $x=180^{\circ}-85^{\circ}=95^{\circ}$.
(b) Since $4x$ and $2x$ are supplementary (linear - pair), $4x + 2x=180^{\circ}$, $6x=180^{\circ}$, $x = 30^{\circ}$. So the angles are $4x = 120^{\circ}$ and $2x = 60^{\circ}$.
(c) Since $3a$, $4a$, and $5a$ are angles around a point, $3a+4a + 5a=360^{\circ}$, $12a=360^{\circ}$, $a = 30^{\circ}$. The angles are $3a = 90^{\circ}$, $4a = 120^{\circ}$, $5a = 150^{\circ}$.

Step6: Solve problem 13

(a) Since $x$ and $x - 44^{\circ}$ are supplementary, $x+(x - 44^{\circ})=180^{\circ}$, $2x-44^{\circ}=180^{\circ}$, $2x=224^{\circ}$, $x = 112^{\circ}$.
(b) Since $x$ and $2x + 24^{\circ}$ are supplementary, $x+(2x + 24^{\circ})=180^{\circ}$, $3x+24^{\circ}=180^{\circ}$, $3x=156^{\circ}$, $x = 52^{\circ}$.
(c) Since $2x$, $x$, and $90^{\circ}$ are angles on a straight - line, $2x+x+90^{\circ}=180^{\circ}$, $3x=90^{\circ}$, $x = 30^{\circ}$.

Step7: Solve problem 14

Since $OP$ and $OQ$ are opposite rays, $x + y=180^{\circ}$.
(a) If $x = 32^{\circ}$, then $y=180^{\circ}-32^{\circ}=148^{\circ}$.
(b) If $y = 24^{\circ}$, then $x=180^{\circ}-24^{\circ}=156^{\circ}$.

Answer:

1. Smaller angle: $68^{\circ}$, Larger angle: $112^{\circ}$
2. (a) No, (b) No, (c) Yes
3. Smaller angle: $27^{\circ}$, Larger angle: $63^{\circ}$
4. (a) $\angle1$ and $\angle3$, $\angle2$ and $\angle4$; (b) $\angle a$ and $\angle d$, $\angle b$ and $\angle e$, $\angle c$ and $\angle f$
5. (a) $x = 95^{\circ}$; (b) $x = 30^{\circ}$, angles are $120^{\circ}$ and $60^{\circ}$; (c) $a = 30^{\circ}$, angles are $90^{\circ}$, $120^{\circ}$, $150^{\circ}$
6. (a) $x = 112^{\circ}$; (b) $x = 52^{\circ}$; (c) $x = 30^{\circ}$
7. (a) $y = 148^{\circ}$; (b) $x = 156^{\circ}$