Certainly! Let's solve the problem step-by-step.
### Step 1: Understand the Problem
We have a triangle whose angles are given in the ratio 1:2:3. We need to find the measures of each angle and then determine the sum of the largest and smallest angles.
### Step 2: Know the Sum of Angles in a Triangle
The sum of the interior angles of any triangle is always 180 degrees.
### Step 3: Set Up the Ratios
Given the ratio of the angles is 1:2:3, we will denote the angles as:
- The smallest angle = [tex]\( 1x \)[/tex]
- The middle angle = [tex]\( 2x \)[/tex]
- The largest angle = [tex]\( 3x \)[/tex]
### Step 4: Create an Equation Based on the Sum of Angles
Since the angles add up to 180 degrees, we can write the following equation:
[tex]\[ 1x + 2x + 3x = 180 \][/tex]
### Step 5: Solve for [tex]\( x \)[/tex]
Combine the terms on the left side:
[tex]\[ 6x = 180 \][/tex]
Now, solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{180}{6} = 30 \][/tex]
### Step 6: Find the Sizes of Each Angle
Using the value of [tex]\( x \)[/tex], we calculate each angle:
- The smallest angle: [tex]\( 1x = 30 \)[/tex] degrees
- The middle angle: [tex]\( 2x = 60 \)[/tex] degrees
- The largest angle: [tex]\( 3x = 90 \)[/tex] degrees
### Step 7: Sum of the Largest and Smallest Angles
We need to find the sum of the largest and smallest angles:
[tex]\[ \text{Sum of the largest and smallest angles} = 90 + 30 = 120 \][/tex] degrees
### Summary
- The smallest angle is 30 degrees.
- The middle angle is 60 degrees.
- The largest angle is 90 degrees.
- The sum of the largest and smallest angles is 120 degrees.
This concludes our solution.
