Geometry Unit 4: How-To Guide

Day 1: Triangle Inequality & Side/Angle Relationships

How to check if a triangle can exist:

  1. Identify the lengths of the three sides.
  2. Add the lengths of the two shortest sides together.
  3. Compare the sum to the length of the longest side.
  4. If the sum is GREATER THAN the longest side, a triangle can be formed.
15 8 10

YES: 8 + 10 > 15

15 6 9

NO: 6 + 9 is not > 15

Worked Example:

Can a triangle have sides 8, 10, and 15?

Step 1 & 2: Shortest sides are 8 and 10. Their sum is \(8 + 10 = 18\).

Step 3 & 4: \(18 > 15\). Since the sum of the two smaller sides is greater than the longest side, YES, it can be a triangle.

How to order sides and angles:

  1. To order sides from shortest to longest, find the angles opposite them and order the angles from smallest to largest.
  2. To order angles from smallest to largest, find the sides opposite them and order the sides from shortest to longest.
105° 30° 45° Shortest Side Medium Side Longest Side

Worked Example:

Order the sides of \( \triangle ABC \) from shortest to longest if \(m\angle A = 45^\circ\) and \(m\angle B = 80^\circ\).

Step 1: Find the third angle: \(180^\circ - 45^\circ - 80^\circ = 55^\circ\). So, \(m\angle C = 55^\circ\).

Step 2: Order angles from smallest to largest: \( \angle A (45^\circ), \angle C (55^\circ), \angle B (80^\circ) \).

Step 3: The sides opposite these angles are \( \overline{BC} \), \( \overline{AB} \), and \( \overline{AC} \). So the order of sides is \( \overline{BC}, \overline{AB}, \overline{AC} \).

Day 2: Triangle Angle Sum & Exterior Angles

How to find a missing interior angle:

  1. Remember that all three angles in a triangle add up to 180°.
  2. Add the two known angles together.
  3. Subtract their sum from 180 to find the third angle.
x° 50° 60°

Worked Example:

Find x if the angles are \(x^\circ, 50^\circ, 60^\circ\).

Equation: \(x + 50 + 60 = 180\)

Solve: \(x + 110 = 180 \implies x = 70\). The missing angle is 70°.

How to find an exterior angle:

  1. Identify the two remote interior angles (the ones not touching the exterior angle).
  2. Add the measures of the two remote interior angles together.
  3. Their sum is equal to the measure of the exterior angle.
40° 90° 130° Remote Interior Angles

Worked Example:

The remote interior angles are \(40^\circ\) and \(90^\circ\). Find the exterior angle.

Equation: Exterior Angle = \(40 + 90\)

Solve: The exterior angle is 130°.

Day 3: Triangle Congruence Postulates

How to determine congruence (SSS, SAS, ASA):

  1. Look at the markings on the two triangles to see which parts are given as congruent.
  2. Count the pairs of congruent Sides (S) and Angles (A).
  3. If you have 3 pairs of Sides, it's SSS.
  4. If you have 2 pairs of Sides and the Angle BETWEEN them, it's SAS.
  5. If you have 2 pairs of Angles and the Side BETWEEN them, it's ASA.

SSS

SAS

ASA

Worked Example:

Two triangles show two pairs of corresponding sides are congruent, and the angle included between those sides is also congruent.

Analysis: We have Side, Angle, Side in that order.

Conclusion: The triangles are congruent by SAS.

Day 4: Congruence with Algebra & Coordinates

How to solve for variables in congruent triangles:

  1. Read the congruence statement (e.g., \( \triangle ABC \cong \triangle DEF \)) to identify corresponding parts.
  2. Set the expressions for corresponding parts equal to each other.
  3. Solve the resulting equation.
CAT AC = 50 DOG DO = 2x+10

Worked Example:

Given \( \triangle CAT \cong \triangle DOG \), \( AC = 50 \), and \( DO = (2x+10) \). Find x.

Step 1: \( \overline{AC} \) corresponds to \( \overline{DO} \).

Step 2: Set up the equation: \( 50 = 2x + 10 \).

Step 3: Solve: \( 40 = 2x \implies x = 20 \). The value of x is 20.

How to prove congruence on the coordinate plane:

  1. Use the Distance Formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \) to find the length of all three sides of the first triangle.
  2. Repeat Step 1 for the second triangle.
  3. Compare the lengths of the corresponding sides. If all three pairs are equal, the triangles are congruent by SSS.

Worked Example:

Is \( \triangle ABC \cong \triangle DEF \)? A(0,0), B(3,0), C(0,4) and D(1,1), E(4,1), F(1,5).

Triangle ABC:
AB = \( \sqrt{(3-0)^2 + (0-0)^2} = \sqrt{9} = 3 \)
BC = \( \sqrt{(0-3)^2 + (4-0)^2} = \sqrt{9+16} = \sqrt{25} = 5 \)
AC = \( \sqrt{(0-0)^2 + (4-0)^2} = \sqrt{16} = 4 \)

Triangle DEF:
DE = \( \sqrt{(4-1)^2 + (1-1)^2} = \sqrt{9} = 3 \)
EF = \( \sqrt{(1-4)^2 + (5-1)^2} = \sqrt{9+16} = \sqrt{25} = 5 \)
DF = \( \sqrt{(1-1)^2 + (5-1)^2} = \sqrt{16} = 4 \)

Conclusion: Since AB=DE, BC=EF, and AC=DF, YES, the triangles are congruent by SSS.