5.1 angles of triangles (pp. 373-384) learning target: prove and use theorems about angles of triangles. classify the triangle by its sides and by measuring its angles. 1. triangle with two marked sides 2. triangle classify △abc by its sides. then determine whether it is a right triangle. 3. a(-2, 3), b(3, 4), c(1, -1) 4. a(2, 3), b(6, 3), c(2, 7) find the measure of the exterior angle. 5. triangle with angles 86° and 46°, exterior angle x° 6. triangle with angles 45° and 5x°, exterior angle (9x + 9)°

Question

Accepted Answer

Let's solve each problem one by one.

### Problem 1: Classify the triangle by its sides and angles (first triangle with two marked sides)
# Explanation:
## Step 1: Classify by sides
The triangle has two sides marked as equal (the tick marks), so by sides, it is an isosceles triangle.
## Step 2: Classify by angles
Since it's isosceles, the base angles are equal. Assuming the unmarked side is the base, the angles opposite the equal sides are equal. If we measure (visually, it looks acute), so it's an acute isosceles triangle.
# Answer: By sides: Isosceles triangle; By angles: Acute isosceles triangle

### Problem 2: Classify the second triangle (with a obtuse angle)
# Explanation:
## Step 1: By angles
The triangle has one angle greater than 90° (the obtuse angle), so it's an obtuse triangle.
## Step 2: By sides
Visually, all sides seem different, so it's a scalene triangle (since no sides are marked equal). So it's an obtuse scalene triangle.
# Answer: By sides: Scalene triangle; By angles: Obtuse scalene triangle

### Problem 3: Classify $ \triangle ABC $ with $ A(-2, 3) $, $ B(3, 4) $, $ C(1, -1) $ by sides and check if right
# Explanation:
## Step 1: Find side lengths using distance formula $ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $
- $ AB $: $ \sqrt{(3 - (-2))^2 + (4 - 3)^2} = \sqrt{5^2 + 1^2} = \sqrt{25 + 1} = \sqrt{26} $
- $ BC $: $ \sqrt{(1 - 3)^2 + (-1 - 4)^2} = \sqrt{(-2)^2 + (-5)^2} = \sqrt{4 + 25} = \sqrt{29} $
- $ AC $: $ \sqrt{(1 - (-2))^2 + (-1 - 3)^2} = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 $

Since all sides $ \sqrt{26} $, $ \sqrt{29} $, $ 5 $ are different, it's scalene by sides.

## Step 2: Check if right triangle (use Pythagorean theorem: $ a^2 + b^2 = c^2 $ for right angle)
Check $ AB^2 + AC^2 $: $ 26 + 25 = 51
eq 29 = BC^2 $
Check $ AB^2 + BC^2 $: $ 26 + 29 = 55
eq 25 = AC^2 $
Check $ AC^2 + BC^2 $: $ 25 + 29 = 54
eq 26 = AB^2 $
So not a right triangle.
# Answer: By sides: Scalene triangle; Not a right triangle

### Problem 4: Classify $ \triangle ABC $ with $ A(2, 3) $, $ B(6, 3) $, $ C(2, 7) $ by sides and check if right
# Explanation:
## Step 1: Find side lengths
- $ AB $: $ \sqrt{(6 - 2)^2 + (3 - 3)^2} = \sqrt{4^2 + 0} = 4 $
- $ BC $: $ \sqrt{(2 - 6)^2 + (7 - 3)^2} = \sqrt{(-4)^2 + 4^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} $
- $ AC $: $ \sqrt{(2 - 2)^2 + (7 - 3)^2} = \sqrt{0 + 4^2} = 4 $

Since $ AB = AC = 4 $, it's isosceles by sides.

## Step 2: Check if right triangle
Check $ AB^2 + AC^2 $: $ 4^2 + 4^2 = 16 + 16 = 32 $ and $ BC^2 = (4\sqrt{2})^2 = 32 $. So $ AB^2 + AC^2 = BC^2 $, so right angle at $ A $.
# Answer: By sides: Isosceles triangle; It is a right triangle (right isosceles triangle)

### Problem 5: Find exterior angle $ x^\circ $
# Explanation:
## Step 1: Recall exterior angle theorem: Exterior angle = sum of two non - adjacent interior angles
The two non - adjacent interior angles are $ 86^\circ $ and $ 46^\circ $.
## Step 2: Calculate $ x $
$ x = 86 + 46 = 132 $
# Answer: $ x = 132^\circ $

### Problem 6: Find $ x $ and exterior angle $ (9x + 9)^\circ $
# Explanation:
## Step 1: Exterior angle theorem: Exterior angle = sum of two non - adjacent interior angles
So $ 9x + 9 = 45 + 5x $
## Step 2: Solve for $ x $
Subtract $ 5x $ and $ 9 $ from both sides: $ 9x - 5x=45 - 9 $
$ 4x = 36 $
$ x = 9 $
## Step 3: Find exterior angle
Substitute $ x = 9 $ into $ 9x + 9 $: $ 9(9)+9 = 81 + 9 = 90 $
# Answer: $ x = 9 $, Exterior angle $ = 90^\circ $

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 1: Classify the triangle by its sides and angles (first triangle with two marked sides)

Step 1: Classify by sides

Step 2: Classify by angles

Step 1: By angles

Step 2: By sides

Step 1: Find side lengths using distance formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)

Step 2: Check if right triangle (use Pythagorean theorem: \( a^2 + b^2 = c^2 \) for right angle)

Answer:

Problem 2: Classify the second triangle (with a obtuse angle)