Chapter 5 Geometry Notes

5.1 Midsegments of Triangles

Note: Midsegments are also ____________________________ to the third side of the triangle.

Identify the 3 midsegments in the diagram.

Identify three pairs of parallel segments in the diagram.

1.) AB II _____

2.) BC II _____

3.) AC II _____

Write an equation to model this theorem based on the figure.

Points J, K, and L are the midpoints of the sides of triangle XYZ. J is the midpoint of XY, K is the midpoint of YZ and L is the midpoint of XZ. Draw a diagram to model the situation with midsegments. Then let XY = 14, JK = 8, and KZ = 5.

4.) Find LK

5.) Find YK

6.) Find XZ

7.) Find XL

8.) Find JL

9.) Find YJ

Finding the values of variables.

Remember that the midsegment is always half of the parallel side.

Always set up an equation and fill in what you have and then solve.

Find the value of x.

10.)

11.)

12.)

X is the midpoint of MN. Y is the midpoint of ON.

13.) Find XZ

14.) If XY = 10, find MO.

15.) If m<M is 64, find m<Y.

5.2 Perpendicular and Angle Bisectors

What have we learned to prove that AP = BP if the segments were drawn in?

What are the ways to prove that a figure is the perpendicular bisector of a segment?

Use the figure at the right.

1.) What is the value of x?

2.) Find AB

3.) Find BC

4.) Find n.

Diagram:

Use the figure at the right.

5.) How far is M from KL?

6.) How far is M from JK?

7.) How is KM related to <JKL?

8.) Find the value of x.

9.) Find m<MKL.

10.) Find m<JMK and m<LMK.

Find the indicated measures.

11.) X, BA, BC

12.) X, EH, EF

13.) X, IK

14.) X, m<UWV, m<UWT

5.3 Bisectors in Triangles

The perpendicular bisectors of a triangle are concurrent at a point known as the ____________________________ and equidistant from the ____________________________.

What point is the circumcenter in the figure?

What is true about PB, PA, and PC?

We could circumscribe a circle about the triangle using what point as the center?

Circumcenters can be inside, outside, or on the triangle itself!

Acute Right Obtuse

What are the coordinates of the circumcenter of a triangle with vertices A(2,7), B(10,7), and C(10,3)?

The angle bisectors of a triangle are concurrent at a point known as the ________________________________ and equidistant from the ________________________.

What point is the incenter?

What is true about PX, PY, and PZ?

We could inscribe a circle inside the triangle using what point?

Find the circumcenter of the triangle with the given vertices.

2.) P(1,-5), Q(4,-5), R(1,-2)

Name the point of concurrency of the angle bisectors.

3.)

4.)

5.)

Find the value of x.

6.)

7.)

8.)

5.4 Medians and Altitudes

The medians of a triangle are concurrent at a point, known as the ____________________, that is ______________________ the distance from the vertex to the midpoint of the opposite side.

DC = 23¿

EC = 23¿

FC = _____

Finding the Length of a Median

1.) Write an equation like the one above.2.) Fill in the appropriate values and solve.

The altitudes of a triangle are concurrent at a point known as the _________________________________________________________. The orthocenter can be inside, outside, or on the triangle. Draw a triangle for each and then construct the orthocenter.

Identify the special segments that create each point of concurrency.

In triangle XYZ, A is the centroid.

1.) If DZ = 12, find ZA and AD.

2.) If AB = 6, find BY and AY.

Is MN a median, an altitude, or neither? Explain.

3.)

4.)

5.)

Name each segment.

6.) A median in triangle STU

7.) An altitude in triangle STU

8.) A median in triangle SBU

9.) An altitude in triangle CBU

Q is the centroid of triangle JKL. PK = 9x+21y. Write expressions to represent PQ and QK.

5.6 Inequalities in One Triangle

The measure of the exterior angle of a triangle is _______ than the measure of either _______.

Circle the angles that 1 is greater than according to this theorem.

Use the figure to explain why m<2 > m<3.

In triangle ACD, CB = CD, so by the ________________ Theorem, m<2 = m<________________. Since ________________ is an exterior angle of triangle ABD, m<1 > m <________________ by the corollary to the ________________. By using substitution m<________________.

Theorem 5-11

If two sides in a triangle are not congruent, then the longer side lies opposite the ___________.

List the angles of each triangle in order from smallest to largest.

2)

3) Triangle XYZ, where XY = 25, YZ = 11, and XZ = 15.

List the sides of each triangle in order from shortest to longest.

4)

5) Triangle MNO, where m<M = 56, m<N = 108, and m<0 = 16.

The sum of the lengths of any two sides in a triangle is _________ the length of the _________.

Fill in the blanks.

Hint: You only need to check the sum of the 2 smallest lengths and make certain it is larger than the 3rd length given.

Can a triangle have sides with the given lengths? Explain.

6) 10 in., 13 in., 18 in.

7) 6 m, 5 m, 12 m

8) 11 ft, 8 ft, 18 ft

Finding the Length of a Third Side

We are finding a range of values. The 3rd side will be in between 2 numbers.

Set up 3 inequalities using x for the 3rd side. OR Add the two numbers and subtract them.

The lengths of two sides of a triangle are given. Find the range of possible lengths for the third side.

13) 4, 8

14) 13, 8

15) 10, 15

5.7 Inequalities in Two Triangles

Write an inequality relating the given side lengths. If there is not enough information to reach a conclusion, write no conclusion.

1.) AB and CB

2.) JL and MO

3.) ST and BT

Finding a Range of Values for Angles

Expression must be greater or less than other angle. AND Expression must be greater than zero.

Find the range of possible values for each variable.

4.)

5.)

Use the diagram at the right. Complete each comparison with < or >. Then complete the explanation.

6.) M<ACB _____ m<DCE

7.) AB _____ DE

8.) BE _____ CE

Write an inequality relating the given angle measures.

9.) m<M and m<R

10.) m<U and m<

Given: m<MON = 80; O is the midpoint of LN

Prove LM > MN