Angles, Triangles, Transformations

and Proofs Packet 1

• Notes and some practice are included

• Homework will be assigned on a daily basis

Topics Covered:

❖ Vocabulary

❖ Angle relationships

❖ Parallel Lines & Transversals

❖ Triangle Theorems

❖ Transformations

❖ Triangle Congruence postulates

❖ Proofs

Test for the unit is 8/21

Name: _____________

Vocabulary • Adjacent Angles: Angles in the same plane that have a common vertex and a

common side, but no common interior points.

• Alternate Exterior Angles: Alternate exterior angles are pairs of angles formed when a third line (a transversal) crosses two other lines. These angles are on opposite sides of the transversal and are outside the other two lines. When the two other lines are

parallel, the alternate exterior angles are equal. • Alternate Interior Angles: Alternate interior angles are pairs of angles formed when

a third line (a transversal) crosses two other lines. These angles are on opposite sides

of the transversal and are in between the other two lines. When the two other lines are parallel, the alternate interior angles are equal.

• Angle: Angles are created by two distinct rays that share a common endpoint (also known as a vertex). ABC or B denote angles with vertex B.

• Bisector: A bisector divides a segment or angle into two equal parts. • Centroid: The point of concurrency of the medians of a triangle.

• Circumcenter: The point of concurrency of the perpendicular bisectors of the sides of a triangle.

• Coincidental: Two equivalent linear equations overlap when graphed.

• Complementary Angles: Two angles whose sum is 90 degrees. • Congruent: Having the same size, shape and measure. Two figures are congruent if

all of their corresponding measures are equal.

• Congruent Figures: Figures that have the same size and shape. • Corresponding Angles: Angles that have the same relative positions in geometric

figures.

• Corresponding Sides: Sides that have the same relative positions in geometric figures • Dilation: Transformation that changes the size of a figure, but not the shape. • Endpoints: The points at an end of a line segment

• Equiangular: The property of a polygon whose angles are all congruent. • Equilateral: The property of a polygon whose sides are all congruent.

• Exterior Angle of a Polygon: an angle that forms a linear pair with one of the angles of the polygon.

• Incenter: The point of concurrency of the bisectors of the angles of a triangle.

• Intersecting Lines: Two lines in a plane that cross each other. Unless two lines are coincidental, parallel, or skew, they will intersect at one point.

• Intersection: The point at which two or more lines intersect or cross.

• Line: One of the basic undefined terms of geometry. Traditionally thought of as a set of

points that has no thickness but its length goes on forever in two opposite directions. AB denotes a line that passes through point A and B.

• Line Segment or Segment: The part of a line between two points on the line. AB denotes a line segment between the points A and B.

• Linear Pair: Adjacent, supplementary angles. Excluding their common side, a linear pair forms a straight line.

• Measure of each Interior Angle of a Regular n-gon: n

n )2(180

• Orthocenter: The point of concurrency of the altitudes of a triangle.

• Parallel Lines: Two lines are parallel if they lie in the same plane and they do not intersect.

• Perpendicular Lines: Two lines are perpendicular if they intersect at a right angle.

• Plane: One of the basic undefined terms of geometry. Traditionally thought of as going on forever in all directions (in two-dimensions) and is flat (i.e., it has no thickness).

• Point: One of the basic undefined terms of geometry. Traditionally thought of as having no

length, width, or thickness, and often a dot is used to represent it. • Proportion: An equation which states that two ratios are equal.

• Ratio: Comparison of two quantities by division and may be written as r/s, r:s, or r to s. • Ray: A ray begins at a point and goes on forever in one direction.

• Reflection: A transformation that "flips" a figure over a line of reflection • Reflection Line: A line that is the perpendicular bisector of the segment with

endpoints at a pre-image point and the image of that point after a reflection.

• Regular Polygon: A polygon that is both equilateral and equiangular. • Remote Interior Angles of a Triangle: the two angles non-adjacent to the exterior

angle.

• Rotation: A transformation that turns a figure about a fixed point through a given angle and a given direction.

• Same-Side Interior Angles: Pairs of angles formed when a third line (a transversal)

crosses two other lines. These angles are on the same side of the transversal and are between the other two lines. When the two other lines are parallel, same-side interior angles are supplementary.

• Same-Side Exterior Angles: Pairs of angles formed when a third line (a transversal) crosses two other lines. These angles are on the same side of the transversal and are outside the other two lines. When the two other lines are parallel, same-side exterior

angles are supplementary. • Scale Factor: The ratio of any two corresponding lengths of the sides of two similar

figures.

• Similar Figures: Figures that have the same shape but not necessarily the same size. • Skew Lines: Two lines that do not lie in the same plane (therefore, they cannot be

parallel or intersect). • Sum of the Measures of the Interior Angles of a Convex Polygon: 180º(n – 2). • Supplementary Angles: Two angles whose sum is 180 degrees.

• Transformation: The mapping, or movement, of all the points of a figure in a plane according to a common operation.

• Translation: A transformation that "slides" each point of a figure the same distance in the

same direction • Transversal: A line that crosses two or more lines. • Vertical Angles: Two nonadjacent angles formed by intersecting lines or segments. Also

called opposite angles.

Name: _________________________________________ Date: _______________________

Complementary, Supplementary, Linear Pairs & Vertical Angles

Geometry Symbols:

• Angle

• Degree

• Right Angle

• Perpendicular

• Segment AB

• Ray CD

• Line EF

• Measure

Naming Angles:

• Name this angle 4 different ways:

Naming Segments

• What is the name of the top side:

Angle Bisector

• Cuts an angle into TWO congruent angles.

• Example: Solve for x:

Linear Pair

• Two angles that are side-by-side, share a common vertex, share a common ray, &

create a straight line.

• EQUATION: ____________________________________

• Examples: Solve for x:

o o

Supplementary Angles

• Two angles that add up to ________. • EQUATION: _______________________

• Examples: Solve for x:

o o

o x and y are supplementary angles. mx = 47. Find my.

o One of two supplementary angles is 46 degrees more than its supplement.

Find the measure of both angles.

Complementary Angles

• Two angles that add up to ________. • EQUATION: _______________________

• Examples: Solve for x:

o o

o One of two complementary angles is 16 degrees less than its complement.

Find the measure of both angles.

Vertical Angles

• 2 angles that share a common vertex & their sides form two pairs of opposite rays.

• EQUATION: ____________________________________

• Examples: Solve for x:

o o o

105o 2x+5 3x+11 119-x

x

70o

2x

x+30

x

y

z

130

2x-17

x+135

x -12 122

Name: ________________________________________________ Date: _______________________

Complementary and Supplementary Angles – Notes

TYPES OF ANGLES:

Sketch:

1. Acute: Acute angles have measures between 0° and 90°.

2. Right: A right angle has measure equal to 90°.

3. Obtuse: Obtuse angles have measures between 90° and 180°.

SPECIAL PAIRS OF ANGLES:

1. Complementary Angles: Pair of angles whose sum of measures equals 90°.

40° and 50° angles are complementary angles because 40° + 50° = 90°.

Example: A 40° angle is called the complement of the 50° angle.

Similarly, the 50° angle is the complement of the 40° angle.

Practice: Find the complement of each angle.

a) 35° b) 48° c) 12°

2. Supplementary Angle: Pair of angles whose sum of measures equals 180°.

60° and 120° angles are complementary angles because 60° + 120° = 180°.

Example: A 60° angle is called the supplement of the 120° angle.

Similarly, the 120° angle is the supplement of the 60° angle.

Practice: Find the supplement of each angle.

a) 40° b) 126° c) 72°

Can you think of a way to remember the difference between complementary

and supplementary angles?

3. Angle Bisector: A ray (or line or segment) that divides an angle into two

congruent angles (two angles with equal measure).

Alternate Interior Angles

•Opposite sides of the

transversal & inside the

parallels

•Are congruent

Equation:

angle = angle

Consecutive Interior Angles

•Same side of the transversal & inside the parallels

•Are supplementary

Equation:

angle + angle = 180

Notes

Isosceles and Equilateral Triangles

You can use these theorems to find angle measures in isosceles triangles.

Find mE in DEF.

mD mE Isosc. Thm.

5x8 (3x14)8 Substitute the given values.

2x 14 Subtract 3x from both sides.

x 7 Divide both sides by 2.

Thus mE 3(7)14 358.

Find each angle measure.

1. mC _____________________ 2. mQ _____________________

3. mH _____________________ 4. mM _____________________

Theorem Examples

Isosceles Triangle Theorem If two sides of a triangle are congruent, then the angles opposite the sides are congruent.

If ,RT RS then /T > /S.

Converse of Isosceles Triangle Theorem If two angles of a triangle are congruent, then the sides opposite those angles are congruent.

If /N > /M, then .LN LM

Notes

Isosceles and Equilateral Triangles continued

You can use these theorems to find values in equilateral triangles.

Find x in STV.

STV is equiangular. Equilateral equiangular

(7x4)8 60 The measure of each of an

equiangular is 60.

7x 56 Subtract 4 from both sides.

x 8 Divide both sides by 7.

Find each value.

5. n _____________________ 6. x _____________________

7. VT _____________________ 8. MN _____________________

Equilateral Triangle Corollary

If a triangle is equilateral, then it is equiangular.

(equilateral equiangular )

Equiangular Triangle Corollary

If a triangle is equiangular, then it is equilateral.

(equiangular equilateral )

If /A > /B > /C, then AB BC CA .

TRANSLATIONSLIDE

Translate right (x + #, y)

Translate left (x – #, y)

Translate up (x, y + #)

Translate down (x, y – #)

REFLECTIONFLIP

Across x-axis ( x , -y ) Change the sign of y

Across y-axis ( -x , y ) Change the sign of x

Across y = x ( y , x ) Swap both

Across y = -x ( -y , -x ) Swap & Negate

ROTATIONTURN

90 CW & 270 CCW ( y , -x ) Negate x & swap

90 CCW & 270 CW ( -y , x ) Negate y & swap

180 either way ( -x , -y ) Negate both

TRA

NSFO

RM

ATI

ON

S

Proving Triangles Congruent

Angle-Side-Angle (ASA) postulate

Angle-Angle-Side (AAS) theorem

Side-Side-Side (SSS) post

Side-Angle-Side post. (SAS)

Triangle Congruence Postulates

Today’s Question: What does it mean for two triangles to be congruent? (MCC9-12.G.SRT5,

MCC9-12.G.CO.7-8)

Congruent Triangles

• ______________________________________________________________________________

• ______________________________________________________________________________

Side – Side – Side (SSS) Congruence Postulate

three sides of one triangle are congruent

to three sides of a second triangle

Side – Angle – Side (SAS) Congruence Postulate

two sides and the included angle of one

triangle are congruent to two sides and

the included angle of a second triangle

Angle – Side – Angle (ASA) Congruence Postulate

two angles and the included side of one

triangle are congruent to two angles and

the included side of a second triangle

Angle – Angle – Side (AAS) Congruence Postulate

two angles and a non-included side of one

triangle are congruent to two angles and a

non-included side of a second triangle

Hypotenuse – Leg (HL) Congruence Postulate

In a right triangle, the hypotenuse and one

leg is congruent to the hypotenuse and leg

of another right triangle

Practice

In each problem, determine if each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If

they are, complete the congruence statement too. If none of these methods work based on

the information given, write “none”. If congruent, finish the congruence statement.

1.

___________

________BIG

2.

___________

________SML

3.

___________

_______OPN

4.

___________

________FLP

5.

___________

________HOT

6.

___________

________CLD

7.

___________

________CAT

8.

___________

________HIP

9.

___________

________PAT

Euclidean Geometry Name ___________________

Triangle Proof Tips

General Directions:

1. You need 3 congruencies to prove that 2 triangles are congruent.

2. You cannot make up your own "given" information.

3. Every time you get a side or angle congruence, move to the next piece of given

information.

4. When you run out of given information, try vertical angles or reflexive property.

❖ If you are given a midpoint of a segment, look for 2 congruent segments.

❖ If you are given an angle bisector, look for two congruent angles.

❖ If you are given parallel lines, look for Alternate Interior angles to be congruent.

❖ If you are given for perpendicular lines look for right angles.

❖ You can use the Reflexive Property and Vertical Angles without given information.

Name: _______________________________________ Date: ______

Matching: Use the choices listed at the bottom in the box for problems #1 – 4

Problem 1:

Statement Reason

1. LM LO 1. Given

2. MN ON 2. Given

3. LN LN 3.

4. LMN LON 4.

Problem 2:

Statement Reason

1. QS RT 1. Given

2. <R and <S are

right <s

2. Given

3. R S 3.

4. 1 2 4.

5. QT QT 5.

6. QST TRQ 6.

Problem 3:

Statement Reason

1. GI KI 1. Given

2. HI JI 2. Given

3. GIH KIJ 3.

4. GIH KIJ 4.

Problem 4:

Statement Reason

1. ,AC BD AB CD 1. Given

2. 1 4, 2 3 2.

3. AD AD 3.

4. ADC DAB 4.

Choices for problems #1 – 4 (some will be used more than once):

AAS

ASA

Alternate Interior Angles are

Given

Reflexive Property

SAS

SSS

Vertical Angles are

Fill in the blank proofs:

Problem 5:

Statement Reason

1. I K 1. Given

2. IHJ KJH 2. Given

3. HJ HJ 3.

4. HJK JHI 4.

Problem 6:

Statement Reason

1. MLN ONL 1. Given

2. _____OLN 2. Given

3. 3. Reflexive Property

4. LNO NLM 4.

Problem 7:

Statement Reason

1. PQ QS 1. Given

2. TQ QP 2. Given

3. PQT RQS 3.

4. PQT SQR 4.

Problem 8:

Statement Reason

1. UV UX 1. Given

2. <UWV and <UWX

are right <s

2. Given

3. 3. Right Angle Congruence

4. 4. Reflexive Property

5. V X 5. Given

6. UWV UWX 6.

Problem 9:

Statement Reason

1. Y C 1. Given

2. YA AC 2. Given

3. 3. Vertical Angles are congruent

4. YZA CBA 4.

Problem 10:

Statement Reason

1. BAC DCA 1. Given

2. AB DC 2. Given

3. 3.

4. ABC CDA 4.

Problem 11:

Statement Reason

1. F I 1. Given

2. E H 2. Given

3. EG HJ 3.

4. EFG HIJ 4.

Problem 12:

Statement Reason

1. O M 1. Given

2. OL LM 2. Given

3. 3.

4. KLO NLM 4.

5. K N 5. CPCTC

Problem 13:

Statement Reason

1. P R 1. Given

2. PSQ RQS 2. Given

3. 3. Reflexive

4. PQS RSQ 4.

Problem 14:

Statement Reason

1. AC BD 1. Given

2. B C 2. Given

3. CAD BDA 3.

4. 4. Reflexive Property

5. ______ACD 5.