angle theorems - handout
date
1. determine the measure of the missing angles.
include the degree symbol in your answers.
a)

a =
b)

b =
c)

c =
d)

d =
e)

e =
f)

f =
g)

g =
h)

h =

k =
i)

m =

n =
2. state the missing values. include a justification for your answer.
a)

b)

c)

value justification using geometric properties
a =
b =
c =
d =

Question

angle theorems - handout
date
1. determine the measure of the missing angles.
include the degree symbol in your answers.
a)

a =
b)

b =
c)

c =
d)

d =
e)

e =
f)

f =
g)

g =
h)

h =

k =
i)

m =

n =
2. state the missing values. include a justification for your answer.
a)

b)

c)

value	justification using geometric properties
a =
b =
c =
d =

Accepted Answer

### Part 1: Determine the measure of the missing angles

#### a)
# Explanation:
## Step1: Angles on a straight line sum to $ 180^\circ $.
$ a + 105^\circ = 180^\circ $
## Step2: Solve for $ a $.
$ a = 180^\circ - 105^\circ = 75^\circ $
# Answer: $ 75^\circ $

#### b)
# Explanation:
## Step1: The right angle is $ 90^\circ $, so $ b + 51^\circ = 90^\circ $.
$ b + 51^\circ = 90^\circ $
## Step2: Solve for $ b $.
$ b = 90^\circ - 51^\circ = 39^\circ $
# Answer: $ 39^\circ $

#### c)
# Explanation:
## Step1: Vertical angles are equal, and angles on a straight line sum to $ 180^\circ $. The angle adjacent to $ 120^\circ $ is $ 60^\circ $, but $ c $ is vertical to the angle supplementary to $ 120^\circ $? Wait, no—actually, $ c $ and the $ 120^\circ $ angle: wait, no, when two lines intersect, vertical angles are equal. Wait, the straight line: $ 120^\circ + $ adjacent angle $ = 180^\circ $, so adjacent angle is $ 60^\circ $, but $ c $ is vertical to the $ 120^\circ $ angle? Wait, no, looking at the diagram (intersecting lines), so $ c $ and the $ 120^\circ $ angle—wait, no, if one angle is $ 120^\circ $, the vertical angle to it would be $ 120^\circ $? Wait, no, when two lines intersect, opposite angles (vertical angles) are equal, and adjacent angles are supplementary. Wait, maybe the diagram is two intersecting lines, so $ c $ is vertical to the angle that's supplementary to $ 120^\circ $? No, wait, let's re-express: if one angle is $ 120^\circ $, the angle adjacent to it (on the straight line) is $ 60^\circ $, but $ c $ is opposite to the $ 120^\circ $ angle? Wait, no, maybe I misread. Wait, the problem says "c" with a line crossing, so two intersecting lines: so $ c $ and the $ 120^\circ $ angle—wait, no, vertical angles are equal. Wait, maybe the $ 120^\circ $ and $ c $ are vertical angles? No, that can't be. Wait, no, when two lines intersect, the sum of adjacent angles is $ 180^\circ $, and vertical angles are equal. So if one angle is $ 120^\circ $, its vertical angle is also $ 120^\circ $, and the adjacent angles are $ 60^\circ $. Wait, maybe the diagram is such that $ c $ is equal to $ 120^\circ $? Wait, no, let's think again. Wait, the user's diagram: "c)" with a horizontal line and a slanted line crossing, making a $ 120^\circ $ angle. So the angle opposite to $ 120^\circ $ (vertical angle) is $ c $, so $ c = 120^\circ $? Wait, no, that would mean adjacent angles are $ 60^\circ $, but vertical angles are equal. Wait, maybe I made a mistake. Wait, no—if two lines intersect, vertical angles are equal. So if one angle is $ 120^\circ $, the vertical angle (c) is also $ 120^\circ $. Wait, but let's check: $ 120^\circ + 60^\circ = 180^\circ $, so adjacent angles are $ 60^\circ $, but vertical angles are equal. So $ c = 120^\circ $.
# Explanation:
## Step1: Vertical angles are equal.
$ c = 120^\circ $ (since they are vertical angles)
# Answer: $ 120^\circ $

#### d)
# Explanation:
## Step1: Sum of angles in a triangle is $ 180^\circ $. So $ d + 73^\circ + 67^\circ = 180^\circ $.
$ d + 73^\circ + 67^\circ = 180^\circ $
## Step2: Simplify and solve for $ d $.
$ d + 140^\circ = 180^\circ $
$ d = 180^\circ - 140^\circ = 40^\circ $
# Answer: $ 40^\circ $

#### e)
# Explanation:
## Step1: The triangle is isosceles (two sides equal), so the base angles are equal? Wait, no, the given angle is $ 70^\circ $, and the other two angles (e and the other) are equal? Wait, the diagram shows two equal sides, so it's an isosceles triangle with vertex angle $ 70^\circ $. So sum of angles: $ 70^\circ + e + e = 180^\circ $ (since the two equal angles are e? Wait, no, the marked sides are the legs, so the base angles are equal. Wait, the vertex angle is $ 70^\circ $, so the two base angles (e) are equal. So $ 70^\circ + 2e = 180^\circ $.
## Step2: Solve for $ e $.
$ 2e = 180^\circ - 70^\circ = 110^\circ $
$ e = \frac{110^\circ}{2} = 55^\circ $
# Answer: $ 55^\circ $

#### f)
# Explanation:
## Step1: The exterior angle is $ 102^\circ $, which is equal to the sum of the two non-adjacent interior angles. So $ 102^\circ = 45^\circ + f $.
$ 102^\circ = 45^\circ + f $
## Step2: Solve for $ f $.
$ f = 102^\circ - 45^\circ = 57^\circ $
# Answer: $ 57^\circ $

#### g)
# Explanation:
## Step1: Angles on a straight line sum to $ 180^\circ $, so $ 60^\circ + g + 75^\circ = 180^\circ $.
$ 60^\circ + g + 75^\circ = 180^\circ $
## Step2: Simplify and solve for $ g $.
$ g + 135^\circ = 180^\circ $
$ g = 180^\circ - 135^\circ = 45^\circ $
# Answer: $ 45^\circ $

#### h)
# Explanation:
## Step1: The triangle is isosceles (marked sides), so $ h = k $, and sum of angles in a triangle is $ 180^\circ $. So $ 75^\circ + h + k = 180^\circ $, and $ h = k $, so $ 75^\circ + 2h = 180^\circ $.
$ 75^\circ + 2h = 180^\circ $
## Step2: Solve for $ h $.
$ 2h = 180^\circ - 75^\circ = 105^\circ $
$ h = \frac{105^\circ}{2} = 52.5^\circ $
## Step3: Since $ h = k $, $ k = 52.5^\circ $
# Answer: $ h = 52.5^\circ $, $ k = 52.5^\circ $

#### i)
# Explanation:
## Step1: Vertical angles are equal, and angles on a straight line sum to $ 180^\circ $. So $ m $ is vertical to the $ 34^\circ $ angle? Wait, no—two intersecting lines: $ 34^\circ $ and $ m $ are vertical angles? Wait, no, if one angle is $ 34^\circ $, its vertical angle is $ m $, so $ m = 34^\circ $. Then $ n $ is supplementary to $ 34^\circ $, so $ n = 180^\circ - 34^\circ = 146^\circ $. Wait, no: when two lines intersect, vertical angles are equal, so $ m = 34^\circ $, and $ n $ is vertical to the angle supplementary to $ 34^\circ $? Wait, no, let's clarify: two intersecting lines, so angle $ 34^\circ $ and $ m $ are vertical angles (equal), so $ m = 34^\circ $. Then $ n $ is adjacent to $ 34^\circ $, so $ n + 34^\circ = 180^\circ $, so $ n = 146^\circ $.
## Step1: Vertical angles: $ m = 34^\circ $
## Step2: Supplementary angles: $ n = 180^\circ - 34^\circ = 146^\circ $
# Answer: $ m = 34^\circ $, $ n = 146^\circ $

### Part 2: State the missing values with justification

#### a) (Triangle with angles $ 64^\circ $, $ 57^\circ $, and $ a $)
# Explanation:
## Step1: Sum of angles in a triangle is $ 180^\circ $. So $ 64^\circ + 57^\circ + a = 180^\circ $.
$ 64^\circ + 57^\circ + a = 180^\circ $
## Step2: Simplify: $ 121^\circ + a = 180^\circ $
## Step3: Solve for $ a $: $ a = 180^\circ - 121^\circ = 59^\circ $
## Justification: Triangle Angle Sum Theorem (sum of interior angles of a triangle is $ 180^\circ $).
# Answer: $ a = 59^\circ $; Justification: Triangle Angle Sum Theorem.

#### b) (Intersecting lines with angle $ 79^\circ $ and $ b $)
# Explanation:
## Step1: Vertical angles are equal. So $ b = 79^\circ $ (since they are vertical angles formed by intersecting lines).
## Justification: Vertical Angles Theorem (vertical angles are congruent).
# Answer: $ b = 79^\circ $; Justification: Vertical Angles Theorem.

#### c) (Triangle with angles $ 41^\circ $, $ 82^\circ $, $ d $, and exterior angle $ c $)
# Explanation:
## Step1: Sum of angles in a triangle: $ 41^\circ + 82^\circ + d = 180^\circ $
$ 41^\circ + 82^\circ + d = 180^\circ $
$ 123^\circ + d = 180^\circ $
$ d = 180^\circ - 123^\circ = 57^\circ $
## Step2: Exterior angle $ c $ is supplementary to $ d $ (linear pair), so $ c + d = 180^\circ $
$ c + 57^\circ = 180^\circ $
$ c = 180^\circ - 57^\circ = 123^\circ $
## Justification for $ d $: Triangle Angle Sum Theorem. Justification for $ c $: Linear Pair Postulate (adjacent angles on a straight line sum to $ 180^\circ $).
# Answer: $ d = 57^\circ $ (Justification: Triangle Angle Sum Theorem); $ c = 123^\circ $ (Justification: Linear Pair Postulate).

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Part 1: Determine the measure of the missing angles

a)

Step1: Angles on a straight line sum to \( 180^\circ \).

Step2: Solve for \( a \).

Step1: The right angle is \( 90^\circ \), so \( b + 51^\circ = 90^\circ \).

Step2: Solve for \( b \).

Answer:

b)