# Unit 5: Triangle Properties

**How To Find the Median in any Triangle**

**Medians** are segments that connect a vertex to the midpoint of the opposite side.

**Step 1:**

Look at what the question is asking you and find those two points in the triangle.

**Step 2:**

Find the vertex and draw a line to the opposite side midpoint.

**Determine the value of ***x* and *AB *if *BQ* is a median of triangle *ABC*, *AQ = 5x - 4, QC = 3x + 12, and AB = 4x - 11.*

**Example of Finding Median**

**AQ = QC**

**5x-4 = 3x+12**

**5x = 3x+18**

**2x = 18**

**x = 9**

**AB = 4(9) - 11**

**AB = 36 - 11**

**AB = 25**

**How To Find the Altitude in any Triangle**

**Altitudes** are segments that are formed by drawing a segment from one side of the triangle to the opposite vertex. It is the shortest distance from to the vertex to the opposite side.

**Step 1: **

Find the vertex that was given to you.

**Step 2:**

Draw a line all the wat across to the opposite side to make a 90 degree angle.

# Example of Finding Alttitude

*SR* is an altitude of *triangle RST. *Solve for *x *if *m<SQR = 5x - 3 *and *m<TRQ = 4x + 3*

m<SRQ + m<TRQ = 90

5x-3 + 4x+3 = 90

5x+4x-3+3 = 90

9x+0 = 90

9x + 90

x = 10

**How To Find an Angle Bisector in any Triangle**

**Angle Bisectors** are segments/rays/lines that bisects and angle of the triangle.

**Step 1:**

Find the angle that is given.

**Step 2:**

Draw a line that starts at the angle and goes through the opposite side.

**m<KJL = m<IJL**

**x+66 = 9x+2**

**66 = 8x +2**

**64 = 8x**

**8 = x**

# Example of Finding Angle Bisector

*JL* is an angle bisector of *triangle IJK*. *m<KJL* = x+66, *m<IJL* = 9x+2, and *KJ* = 2x-5. Determine *x* and *KJ*.

**KJ = 2(8)-5**

**KJ = 16-5**

**KJ = 11**

# Perpendicular Bisector Property

**How to Find a Perpendicular Bisector in any Triangle**

**Perpendicular bisectors** are segments/rays/lines that pass through the triangle and are perpendicular to one side of the triangle.

**Step 1: **

Find the midpoint from the segment that you were given.

**Step 2:**

Draw a line from that midpoint straight up until you pass another side.

**Step 3:**

Draw a square at the bottom corner because you just made a 90 degree angle.

*XY* is a perpendicular bisector of *PR*. Determine *PR* of *m<PXY* = 8x+2, and *PX* = 3x-7.

# Example of Finding Perpendicular Bisector

# Triangle Inequality Theorem

**Triangle Inequality Theorem:** the sum of any two sides of a triangle must be greater than the third.

32 + 51 > 20 *

32 + 20 > 51 *

20 + 51 > 32 *

AB + BC > AC

AB + AC > BC

AC + BC > AB

1 + 2 > 3

1 + 3 > 2

2 + 3 > 1

9 + 4 > 3 *

9 + 3 > 4 *

3 + 4 > 9 x

# Comparing Angles Using Side Measures and Vice Versa

### The **mid-size** side is opposite the **middle** angles and vice versa.

### The **longest** side is opposite the **biggest** angle and vice versa.

### The **shortest** side is opposite the **smallest** angles and vice versa.

### Order the sides from shortest to longest.

# SSS and SAS Inequality Theorems

**Converse of Hinge Theorem (SSS Inequality):**

Suppose you have two pairs of congruents sides in two different triangles.

**Hinge Theorem (SAS Inequality):**

Start with two pairs of congruent sides in two different triangles.

**Included angles are different measures.**

*Triangle ABC is congruent to Triangle DEF.*

**Included angles are congruent.**

# Inequalities Between 2 Trinangles

**Compare the angles to the non-congruent sides.**