1. Pearson Unit 1 Topic 3: Parallel & Perpendicular Lines 3-5: Parallel Lines and Triangles Pearson Texas Geometry ©2016 Holt Geometry Texas ©2007

2. TEKS Focus:
(6)(D) Verify theorems about the relationships in triangles, including proof of the Pythagorean Theorem, the sum of interior angles, base angles of isosceles triangles, midsegments, and medians, and apply these relationships to solve problems.
(1)(F) Analyze mathematical relationships to connect and communicate mathematical ideas.
(1)(G) Display, explain, or justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

3. Terms – Classifying by Angles
Vocabulary Term
Definition
Diagram
Three acute angles.
Acute
Equiangular
Three equal angles.
One 90⁰ angle.
Right
Obtuse
One obtuse angle.

4. Terms – Classifying by Sides
Vocabulary Term
Definition
Picture
Equilateral
Three congruent sides.
At least two congruent sides.
Isosceles
Scalene
No congruent sides.

5. An interior angle is formed by two sides of a triangle
An interior angle is formed by two sides of a triangle. An exterior angle is formed by one side of the triangle and extension of an adjacent side.
4 is an
exterior angle.
Exterior
Interior
3 is an interior angle.

6. The remote interior angles of 4 are 1 and 2.
Each exterior angle has two remote interior angles. The 2 remote interior angles of a triangle are the two nonadjacent interior angles corresponding to each exterior angle of the triangle.
4 is an exterior angle.
The remote interior angles
of 4 are 1
and 2.
Exterior
Interior
3 is an interior angle.

7. An auxiliary line is a line that you can add to a diagram to help explain relationships in proofs.

8. Investigation: Angle Relationships in a Triangle
Sketchpad
Straightedge and protractor

10. The Converse of the Corresponding Angles Postulate is used to construct parallel lines. The Parallel Postulate guarantees that for any line ℓ, you can always construct a parallel line through a point that is not on ℓ.
Note

13. Example: 1 After an accident, the positions of
cars are measured by law enforcement
to investigate the collision. Use the diagram drawn from the information collected to find mXYZ.
mXYZ + mYZX + mZXY = 180°
Sum. Thm
Substitute 40 for mYZX and 62 for mZXY.
mXYZ = 180
mXYZ = 180
Simplify.
mXYZ = 78°
Subtract 102 from both sides.

14. Example: 2 Lin. Pair Thm. and  Add. Post. Substitute 62 for mYXZ.
After an accident, the positions of cars are measured by law enforcement to investigate
the collision. Use the diagram drawn from
the information collected to find mYWZ.
Step 1 Find mWXY.
mYXZ + mWXY = 180°
Lin. Pair Thm. and  Add. Post.
62 + mWXY = 180
Substitute 62 for mYXZ.
mWXY = 118°
Subtract 62 from both sides.
Step 2 Find mYWZ.
mYWX + mWXY + mXYW = 180°
Sum. Thm
Substitute 118 for mWXY and 12 for mXYW.
mYWX = 180
mYWX = 180
Simplify.
mYWX = 50°
Subtract 130 from both sides.
mYWX = m YWZ =50°

15. Example: 3 Use the diagram to find mMJK. mMJK + mJKM + mKMJ = 180°
Sum. Thm
Substitute 104 for mJKM and 44 for mKMJ.
mMJK = 180
mMJK = 180
Simplify.
mMJK = 32°
Subtract 148 from both sides.

16. Example: 4 One of the acute angles in a right triangle measures 2x°.
What is the measure of the other acute angle?
Let the acute angles be A and B, with mA = 2x°.
mA + mB = 90°
Acute s of rt. are comp.
2x + mB = 90
Substitute 2x for mA.
mB = (90 – 2x)°
Subtract 2x from both sides.

17. Example: 5 Find mB. mB = 2x + 3 = 2(26) + 3 = 55° Ext.  Thm.
mA + mB = mBCD
Ext.  Thm.
Substitute 15 for mA, 2x + 3 for mB, and 5x – 60 for mBCD.
15 + 2x + 3 = 5x – 60
2x + 18 = 5x – 60
Simplify.
Subtract 2x and add 60 to both sides.
78 = 3x
26 = x
Divide by 3.
mB = 2x + 3 = 2(26) + 3 = 55°

18. Example: 6 Find mACD. mACD = 6z – 9 = 6(25) – 9 = 141° Ext.  Thm.
mACD = mA + mB
Ext.  Thm.
Substitute 6z – 9 for mACD, 2z + 1 for mA, and 90 for mB.
6z – 9 = 2z
6z – 9 = 2z + 91
Simplify.
Subtract 2z and add 9 to both sides.
4z = 100
z = 25
Divide by 4.
mACD = 6z – 9 = 6(25) – 9 = 141°

19. EXTRA EXAMPLES NOT USED IN COMPOSITION BOOK FOLLOW.
ALSO REMEMBER TO LOG-ON TO YOUR PEARSON ACCOUNT TO LOOK AT OTHER EXAMPLES BEFORE BEGINNING THE ON-LINE HW AND THE WRITTEN HW.

20. Example: 7
The measure of one of the acute angles in a right triangle is 63.7°. What is the measure of the other acute angle?
Let the acute angles be A and B, with mA = 63.7°.
mA + mB = 90°
Acute s of rt. are comp.
mB = 90
Substitute 63.7 for mA.
mB = 26.3°
Subtract 63.7 from both sides.

21. Example: 8
The measure of one of the acute angles in a right triangle is x°. What is the measure of the other acute angle?
Let the acute angles be A and B, with mA = x°.
mA + mB = 90°
Acute s of rt. are comp.
x + mB = 90
Substitute x for mA.
mB = (90 – x)°
Subtract x from both sides.

22. Example: 9
The measure of one of the acute angles in a right triangle is What is the measure of the other acute angle?
2° 5
Let the acute angles be A and B, with mA =
2° 5
mA + mB = 90°
Acute s of rt. are comp.
mB = 90
2 5
Substitute for mA.
2 5
mB = 41 3° 5
Subtract from both sides.
2 5