Triangle Inequality & Side/Angle Relationships
Geometry - Unit 4, Day 1
Today's Objective
Do Now: Comparing Sizes
Answer the following questions.
1. Order the numbers from least to greatest: 15, 8, 21.
2. In a right triangle, which side is always the longest?
Triangle Inequality Theorem
The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Shortcut: Just check if the two smallest sides add up to be more than the largest side!
How to Check for a Triangle
Step 1: Identify the three side lengths.
Step 2: Find the two smallest side lengths.
Step 3: Add the two smallest sides together.
Step 4: Compare the sum to the largest side.
- If the sum is GREATER THAN the largest side, it's a triangle.
- If the sum is LESS THAN or EQUAL TO the largest side, it is not a triangle.
Practice: Can it be a triangle?
I Do
Sides: 5, 8, 11
Check: \( 5 + 8 > 11 \)
\( 13 > 11 \). Yes!
✔️ It's a triangle.
We Do
Sides: 4, 12, 7
Smallest two: 4, 7
Check: \( 4 + 7 > 12 \)
\( 11 > 12 \). No!
❌ Not a triangle.
You Do
Sides: 9, 15, 6
Smallest two: 6, 9
Check: \( 6 + 9 > 15 \)
\( 15 > 15 \). No! (Must be greater)
❌ Not a triangle.
I Do: Sides: 5, 8, 11. Is it a triangle? Yes
We Do: Sides: 4, 12, 7. Is it a triangle?
You Do: Sides: 9, 15, 6. Is it a triangle?
Side-Angle Relationship
In any triangle, the longest side is opposite the largest angle, and the shortest side is opposite the smallest angle.
• Side a is opposite ∠A.
• Side b is opposite ∠B.
• Side c is opposite ∠C.
If \( m\angle A > m\angle B \), then \( a > b \).
How to Order Sides and Angles
Given Angles, Find Side Order:
- List the angles from smallest to largest.
- Find the side opposite each angle.
- List the sides in the same order (shortest to longest).
Given Sides, Find Angle Order:
- List the sides from shortest to longest.
- Find the angle opposite each side.
- List the angles in the same order (smallest to largest).
Practice: Ordering Sides & Angles
I Do
In \( \triangle ABC \), angles are:
\(m\angle A = 40^\circ, m\angle B=60^\circ, m\angle C=80^\circ \)
Order sides (shortest to longest): \( \overline{BC}, \overline{AC}, \overline{AB} \)
We Do
In \( \triangle ABC \), sides are:
AB=7, BC=10, AC=9
Order sides: \( \overline{AB}, \overline{AC}, \overline{BC} \)
Order angles (smallest to largest): \( \angle C, \angle B, \angle A \)
You Do
In \( \triangle XYZ \), angles are:
\( m\angle X = 55^\circ, m\angle Y = 35^\circ \)
Third angle: \( m\angle Z = 90^\circ \)
Order angles: \( \angle Y, \angle X, \angle Z \)
Order sides (shortest to longest): \( \overline{XZ}, \overline{YZ}, \overline{XY} \)
I Do: Angles: 40°, 60°, 80°. Order sides shortest to longest: Side opp 40°, Side opp 60°, Side opp 80°
We Do: Sides: 7, 10, 9. Order angles smallest to largest:
You Do: Angles: 55°, 35°, ?. Find the third angle and order sides shortest to longest:
Independent Practice
Level 1 Problems:
1. Can a triangle have side lengths of 10, 12, and 23? Why or why not?
2. In \( \triangle XYZ \), \( m\angle X = 95^\circ \) and \( m\angle Y = 25^\circ \). What is the longest side?
Level 2 Problem:
3. Two sides of a triangle are 8 inches and 13 inches. Find the range of possible lengths for the third side.
Level 3 Problem:
4. List the sides of the triangle below in order from shortest to longest.
\( \triangle ABC \) with \( m\angle A = (3x+10)^\circ \), \( m\angle B = (2x+5)^\circ \), and \( m\angle C = (x-3)^\circ \).
Independent Practice
1. Can a triangle have side lengths of 10, 12, and 23? Why or why not?
2. In \( \triangle XYZ \), \( m\angle X = 95^\circ \) and \( m\angle Y = 25^\circ \). What is the longest side?
3. Two sides of a triangle are 8 inches and 13 inches. Find the range of possible lengths for the third side.
4. List the sides of \( \triangle ABC \) in order from shortest to longest if \( m\angle A = (3x+10)^\circ \), \( m\angle B = (2x+5)^\circ \), and \( m\angle C = (x-3)^\circ \).
Exit Ticket
In \( \triangle PQR \), list the angles in order from smallest to largest if PQ = 14, QR = 18, and PR = 11.
Sides from shortest to longest: PR (11), PQ (14), QR (18)
Angles from smallest to largest: \( \angle Q, \angle R, \angle P \)