e 2 list the angles of each triangle in order from smallest to largest. 90) 4. 5. 6.

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Accepted Answer

### Problem 4: Triangle $ \triangle KLM $ with sides $ KL = 2.7 $, $ KM = 4.3 $, $ LM = 5.8 $
# Explanation:
## Step 1: Recall the relationship between side lengths and angles in a triangle
In a triangle, the larger the side length, the larger the angle opposite to it. So we first identify the sides and their opposite angles.
- Side $ KL = 2.7 $ is opposite angle $ \angle M $
- Side $ KM = 4.3 $ is opposite angle $ \angle L $
- Side $ LM = 5.8 $ is opposite angle $ \angle K $

## Step 2: Order the sides from shortest to longest
The side lengths are $ 2.7 < 4.3 < 5.8 $, so $ KL < KM < LM $

## Step 3: Order the angles opposite these sides
Since the side opposite $ \angle M $ is $ KL = 2.7 $ (shortest), $ \angle M $ is the smallest. The side opposite $ \angle L $ is $ KM = 4.3 $, so $ \angle L $ is next. The side opposite $ \angle K $ is $ LM = 5.8 $ (longest), so $ \angle K $ is the largest.
So the order of angles from smallest to largest is $ \angle M < \angle L < \angle K $

# Answer: $ \angle M, \angle L, \angle K $

### Problem 5: Triangle $ \triangle CDE $ with $ \angle E = 105^\circ $, sides $ CE = x $, $ DE = 3x $
# Explanation:
## Step 1: Recall the angle - sum property of a triangle
The sum of angles in a triangle is $ 180^\circ $. Let $ \angle C=x $, $ \angle D = 3x $ and $ \angle E=105^\circ $
So $ x + 3x+105^\circ=180^\circ $

## Step 2: Solve for $ x $
Combine like terms: $ 4x+105^\circ = 180^\circ $
Subtract $ 105^\circ $ from both sides: $ 4x=180^\circ - 105^\circ=75^\circ $
Divide both sides by 4: $ x=\frac{75^\circ}{4} = 18.75^\circ $
Then $ 3x=3\times18.75^\circ = 56.25^\circ $

## Step 3: Order the angles
We have $ \angle C=x = 18.75^\circ $, $ \angle D = 3x=56.25^\circ $, $ \angle E = 105^\circ $
So the order from smallest to largest is $ \angle C<\angle D<\angle E $

# Answer: $ \angle C, \angle D, \angle E $

### Problem 6: Right - triangle $ \triangle GIH $ with $ \angle I = 90^\circ $, $ GI = 6 $, $ HI = 4 $
# Explanation:
## Step 1: Recall the Pythagorean theorem to find the third side (optional, but we can use side - angle relationship)
First, find the length of $ GH $ using Pythagorean theorem: $ GH=\sqrt{GI^{2}+HI^{2}}=\sqrt{6^{2} + 4^{2}}=\sqrt{36 + 16}=\sqrt{52}\approx7.21 $
Now, identify the sides and their opposite angles:
- Side $ HI = 4 $ is opposite $ \angle G $
- Side $ GI = 6 $ is opposite $ \angle H $
- Side $ GH\approx7.21 $ is opposite $ \angle I = 90^\circ $

## Step 2: Order the sides from shortest to longest
$ 4<6 < 7.21 $, so $ HI<GI < GH $

## Step 3: Order the angles opposite these sides
The side opposite $ \angle G $ is $ HI = 4 $ (shortest), so $ \angle G $ is the smallest. The side opposite $ \angle H $ is $ GI = 6 $, so $ \angle H $ is next. The side opposite $ \angle I $ is $ GH\approx7.21 $ (longest) and $ \angle I = 90^\circ $ is the largest.
So the order of angles from smallest to largest is $ \angle G<\angle H<\angle I $

# Answer: $ \angle G, \angle H, \angle I $

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QUESTION IMAGE

Question

Explanation:

Problem 4: Triangle \( \triangle KLM \) with sides \( KL = 2.7 \), \( KM = 4.3 \), \( LM = 5.8 \)

Step 1: Recall the relationship between side lengths and angles in a triangle

Step 2: Order the sides from shortest to longest

Step 3: Order the angles opposite these sides

Step 1: Recall the angle - sum property of a triangle

Step 2: Solve for \( x \)

Step 3: Order the angles

Step 1: Recall the Pythagorean theorem to find the third side (optional, but we can use side - angle relationship)

Step 2: Order the sides from shortest to longest

Step 3: Order the angles opposite these sides

Answer:

Problem 5: Triangle \( \triangle CDE \) with \( \angle E = 105^\circ \), sides \( CE = x \), \( DE = 3x \)