5.1: Midsegments of Triangles

NOTE: Midsegments are also to the third side in the triangle.

Example: Identify the 3 midsegments in the diagram.

Examples: Identify three pairs of parallel segments in the diagram.

1. AB 2. BC 3. AC

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

Examples: Points J, K, and L are the midpoints of the sides of ΔXYZ. J is the midpoint of , K is the midpoint of and L is the midpoint of . 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.

Algebra Find the value of x.

10. 11. 12.

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

13. Find XZ.

14. If XY = 10, find MO.

15. If mM is 64, find mY.

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?

Examples: Use the figure at the right for Exercises 1–3.

1. What is the value of x?

2. Find AB. 3. Find BC.

Example 4: Find n

Examples: Use the figure at the right for Exercises 5–10.

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 mMKL. 10. Find mJMK and mLMK.

Examples: Find the indicated measures.

11. x, BA, BC 12. x, EH, EF

13. x, IK 14. x, mUWV, mUWT

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?

Example: Find the circumcenter of the triangle with the given vertices.

2) P(1, -5)

Q(4, -5)

R(1, -2)

Examples: Name the point of concurrency of the angle bisectors.

3) 4) 5)

Examples: Find the value of x.

6) 7) 8)

x

y

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.

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.

Examples: In ∆XYZ, A is the centroid.

1. If DZ = 12, find ZA and AD. 2. If AB = 6, find BY

and AY.

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

3. 4. 5.

Examples: In Exercises 6–9, name each segment.

6. a median in ∆STU

7. an altitude in ∆STU

8. a median in ∆SBU

9. an altitude in ∆CBU

10. Q is the centroid of ∆JKL. PK = 9x + 21y.

Write expressions to represent PQ and

QK.

5.5: Indirect Proof

1) What are you trying to prove?

2) What is the opposite of what you are trying to prove?

3) What is the first step of this proof then?

Examples: Complete the first step of an indirect proof of the given statement.

4. There are fewer than 11 pencils in the box.

5. If a number ends in 0, then it is not divisible by 3.

6. If a number ends in x, then it is a multiple of 5.

7. mXYZ < 100

8. ∆DEF is a right triangle.

Identifying Contradictions

Indirect proofs should always lead to a contradiction.

Use your definitions to find a contradiction of two statements.

Examples: Identify the two statements that contradict each other.

9. I. MN || GH

II. MN and GH do not intersect.

III. MN and GH are skew.

10. I. ∆CDE is equilateral.

II. mC and mE have the same measure.

III. mC > 60

11. I. ∆JKL is scalene.

II. ∆JKL is obtuse.

III. ∆JKL is isosceles.

Example: Writing an indirect proof. Supply the missing

reasons.

Complete the indirect proof.

12. Given: S W

T V

Prove: VWTS

Assume temporarily that .

Then by the Converse of the , S and W

cannot be . This contradicts the given information

that . Therefore, TS must be VW .

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.

Example 1: Use the figure to explain why .

In so by the

Theorem, . Since is an exterior

angle of , by the Corollary to the

. By using substitution

.

Theorem 5-11

If two sides in a triangle are note congruent, then the longer side lieas opposite the

.

For Exercises 2–3, list the angles of each triangle in order from smallest to largest.

2. 3. ∆XYZ, where XY = 25, YZ = 11, and

XZ = 15.

For Exercises 4–5, list the sides of each triangle in order from shortest to longest.

4. 5. ∆MNO, where mM = 56, mN = 108, and

mO = 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.

Examples: 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.

Ways to Find:

Set up 3 inequalities using x for the 3rd

side

OR

Add the 2 numbers and subtract them.

Examples:The lengths of two sides of a triangle are given. Find the range of possible lengths

for the third side.

13. 4, 8 (Set up inequalities) 14. 13, 8 15. 10, 15

5.7: Inequalities in Two Triangles

Examples: 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.

Examples: Find the range of possible values for each variable.

4. 5.

Examples: Use the diagram at the right for Exercises 6–8. Complete each comparison with <

or >. Then complete the explanation.

6. mACB mDCE

7. AB DE

8. BE CE

Examples: Write an inequality relating the given angle measures.

9. mM and mR 10. mU and mX