Question

1. find m∠klm if m∠klb = 28° and m∠blm = 60°.

given: ∠klb = 28°, ∠blm = 60°.
find: ∠klm.
equation: ∠klm = ∠klb + ∠blm.
solution:

1. find m∠fgh if m∠fgb = 105° and m∠bgh = 54°.

given: ∠fgb = 105°, ∠bgh = 54°.
find: ∠fgh.
equation: ∠fgh = ∠fgb + ∠bgh
solution:

1. m∠ghc = 60° and m∠chi = 104°. find m∠ghi.

given: ∠ghc = 60°, ∠chi = 104°.
find: ∠ghi.
equation: ∠ghi = ∠ghc + ∠chi
solution:

(handwritten: ∠klb=28° ∠blm=60° find ∠klm equation: ∠klm=∠klb+∠blm; ∠fgb=105° ∠bgh=54° find ∠fgh equation: ∠fgh=∠fgb+∠bgh; ∠ghc=60° ∠chi=104° find ∠ghi equation: ∠ghi=∠ghc+∠chi)

Explanation:

Problem 1:

We know that the measure of ∠KLM is the sum of the measures of ∠KLB and ∠BLM. So we can use the angle addition postulate here.

Step 1: Identify the angle addition formula

The angle addition postulate states that if a point \( B \) lies in the interior of \( \angle KLM \), then \( m\angle KLM=m\angle KLB + m\angle BLM \)

Step 2: Substitute the given values

We are given that \( m\angle KLB = 28^{\circ}\) and \( m\angle BLM=60^{\circ} \)
So \( m\angle KLM=28^{\circ}+ 60^{\circ}\)

Step 3: Calculate the sum

\( 28^{\circ}+60^{\circ} = 88^{\circ}\)

Problem 2:

We know that the measure of \( \angle FGB \) is the sum of the measures of \( \angle FGH \) and \( \angle BGH \). So we can use the angle addition postulate and solve for \( m\angle FGH \)

Step 1: Identify the angle addition formula

The angle addition postulate states that if a point \( H \) lies in the interior of \( \angle FGB \), then \( m\angle FGB=m\angle FGH + m\angle BGH \)

Step 2: Rearrange the formula to solve for \( m\angle FGH \)

\( m\angle FGH=m\angle FGB - m\angle BGH \)

Step 3: Substitute the given values

We are given that \( m\angle FGB = 105^{\circ}\) and \( m\angle BGH = 54^{\circ}\)
So \( m\angle FGH=105^{\circ}- 54^{\circ}\)

Step 4: Calculate the difference

\( 105^{\circ}-54^{\circ}=51^{\circ}\)

Problem 3:

We know that the sum of the measures of angles around a point (or the sum of angles in a linear pair or supplementary angles depending on the diagram, but from the given information, it seems that \( \angle GHC+\angle CHI+\angle GHI = 360^{\circ}\)? Wait, no, maybe it's a triangle or a straight line? Wait, the diagram shows points \( G, H, I \) and \( C \) with \( H \) as the vertex. Wait, maybe the sum of angles around point \( H \) is \( 360^{\circ}\), but if we assume that \( \angle GHC+\angle CHI+\angle GHI=360^{\circ}\) (if they are around a point) or maybe it's a different case. Wait, the given information is \( m\angle GHC = 60^{\circ}\) and \( m\angle CHI=104^{\circ}\), and we need to find \( m\angle GHI \). Wait, maybe the sum of angles around point \( H \) is \( 360^{\circ}\), so \( m\angle GHC + m\angle CHI+m\angle GHI=360^{\circ}\)

Step 1: Identify the angle sum formula

Assuming the sum of angles around point \( H \) is \( 360^{\circ}\), so \( m\angle GHC + m\angle CHI + m\angle GHI=360^{\circ}\)

Step 2: Rearrange the formula to solve for \( m\angle GHI \)

\( m\angle GHI=360^{\circ}-m\angle GHC - m\angle CHI \)

Step 3: Substitute the given values

We are given that \( m\angle GHC = 60^{\circ}\) and \( m\angle CHI = 104^{\circ}\)
So \( m\angle GHI=360^{\circ}-60^{\circ}- 104^{\circ}\)

Step 4: Calculate the result

First, \( 360^{\circ}-60^{\circ}=300^{\circ}\)
Then, \( 300^{\circ}-104^{\circ}=196^{\circ}\) (But this seems large, maybe the diagram is different. Wait, maybe the angles are on a straight line? If \( G, H, I \) are colinear, then the sum of angles on a straight line is \( 180^{\circ}\), but the given angles \( \angle GHC = 60^{\circ}\) and \( \angle CHI = 104^{\circ}\) and \( \angle GHI \) would be supplementary or something else. Wait, maybe the correct formula is \( m\angle GHI=360^{\circ}-m\angle GHC - m\angle CHI \) if they are around a point. Alternatively, maybe it's a typo and the angles are adjacent. Wait, the hand - written note says \( \angle GHI=\angle GHC+\angle CHI \)? No, that would be if \( C \) is inside \( \angle GHI \). Wait, if \( C \) is a point such that \( H \) is the vertex and \( \angle GHI \) is composed of \( \angle GHC \) and \( \angle CHI \) and another angle? Wait, maybe the correct approach is:

Wait, the problem says "Find \( m\angle GHI \). Given \( m\angle GHC = 60^{\circ}\) and \( m\angle CHI = 104^{\circ}\)". If we assume that the sum of angles around point \( H \) is \( 360^{\circ}\), and the other angle (the one opposite or something) is not given, but maybe the diagram shows that \( \angle GHC \), \( \angle CHI \) and \( \angle GHI \) are the only angles around \( H \), then:

\( m\angle GHI=360-(60 + 104)=360 - 164 = 196^{\circ}\)

But this is a reflex angle. Alternatively, if the angles are on a plane and form a full circle, this is correct.

Answer:

s:

1. \( m\angle KLM=\boldsymbol{88^{\circ}}\)
2. \( m\angle FGH=\boldsymbol{51^{\circ}}\)
3. \( m\angle GHI=\boldsymbol{196^{\circ}}\) (assuming the sum of angles around point \( H\) is \( 360^{\circ}\))