Recall that the midpoint is a point on a line or segment that bisects the line or segment. In a triangle, the midsegment is a segment that connects the midpoint of two sides.

LN is ½ of AB, and LN II AB

Example 1:

In triangle ABC, D, E, and F are midpoints. Find CB, DE, and FE. Also, name the parallel segments.

Example 2:

If m<B = 80, find the rest of the angle measures.

m<C=________ m<DEB=_______ m<BDE= ________

Example 3:

CD is a new bridge being built over a lake as shown. Find the length of the bridge.

Coordinate geometry uses the coordinate plane.

Example 4

Find the length of the midsegment RS, and compare its slope to that of OQ.

Chapter 5 Sect. 1

Chapter 5 Sect. 2

Triangles play a key role in relationships involving perpendicular bisectors and angle bisectors. Remember that a bisector cuts something into two equal parts.

Example 1:

CD is the perpendicular bisector of AB. Find CA and DB.

Example 2:

Find the value of x, then find FD and FB.

Example 3:

a. How far is K from EH and from ED?

b. What can you conclude about EK?

c. Find the value of x.

d. Find m< DEH

When two or more lines intersect in one point, they are concurrent. The point at which they intersect is the point of concurrency. In a triangle, the three altitudes, the three medians, the three angle bisectors, and the three perpendicular bisectors are congruent at special points.

Chapter 5 Sect. 3

Example 1:

Complete the following steps to locate the centroid.

a. Find the coordinates of A, B, and C.

b. Use the formula (X + X + X , Y + Y +Y ) to find the centroid.

Example 2:

Find the center of the circle that you can circumscribe about the triangle with vertices (0, 0), (–8, 0), and (0, 6).

Example 3:

The Jacksons want to install the largest possible circular pool in their triangular backyard. Where would the largest possible pool be located? What could they use the figure out where to build the pool?

Example 4:

D is the centroid of ABC and DE = 6. Find BE and BD.

1 2 3 1 2 3

3 3

Chapter 5 Sect. 4

Recall what conditional and converse statements are.

For example:

CONDITIONAL: If it is sunny outside, then we will go to the beach.

CONVERSE: If we go to the beach, then it is sunny outside.

When we negate a phrase, we insert the word “not” into it, or say the opposite of what is stated.

Inverse: negates the hypothesis and conclusion

Contrapositive: the negation of a converse‛s hypothesis and conclusion.

We have used indirect reasoning and proofs before. Now, let‛s put them together to make an indirect proof. A proof involving indirect reasoning is an indirect proof. In an indirect proof, a statement and its negation often are the only possibilities.

For example:

Given: Julie and Jose are holding hands, so they must be dating.

Step 1: Assume Julie and Jose are not dating.

Step 2: If Julie and Jose are not dating, then the couple wouldn‛t be holding hands. This contradicts the statement that they are holding hands.

Step 3: The couple must be dating.

Given: The sum of the angles in a polygon add up to 180 degrees, then the polygon must be a triangle.

Step 1: Assume that the polygon is not a triangle.

Step 2: If the polygon was another figure, such as a rectangle, then the sum of the angles would be greater than 180 degrees. This contradicts the statement that the sum of the angles added up to 180.

Step 3:The figure must be a triangle.

Example 3:

Write the indirect proof for these statements.

a. An obtuse triangle cannot contain a right angle.

Step 1:

Step 2:

Step 3:

b. Given: Triangle ABC is scalene, m< ABX = 36 and m<CBX =36

Prove: XB is not perpendicular to AC.

Step 1:

Step 2:

Step 3:

Chapter 5 Sect. 5

The Comparison Property of Inequality allows you to prove the following corollary to the Exterior Angle Theorem for triangles (Theorem 3-8).

A landscape architect is designing a triangular deck. List the angles of ABC in order from smallest to largest.

List the sides of the TUV in order from shortest to longest.

Not every set of three segments can form a triangle. The lengths of the segments must be related in a certain way.

Notice that only one of the sets of three segments above can form a triangle. The sum of the smallest two lengths must be greater than the greatest length.

a. Can a triangle have sides with lengths 2 m, 7 m, and 9 m?

b. Can a triangle have sides with lengths 4 yd, 6 yd, and 9 yd?

c. A triangle has sides of lengths 3 in. and 12 in. Describe the lengths possible for the third side.

In the diagram, m 2 = m 1 by the Isosceles Triangle Theorem. Explain why

m 2 > m 3.

When you empty a container of juice into two glasses, it is difficult to be sure that the glasses get equal amounts. You can be sure, however, that each glass holds less than the original amount in the container. This is a simple application of the Comparison Property of Inequality.

Example 1:

Example 2:

Example 3:

Example 4: