Download Theorems and Postulates for Using in Proofs

Name:_________________________________________
October 13, 2015
Math2
U1 Test Review
Chapter 6ABC Review
Students should be able to:
 Use invariant vocabulary and various other geometric vocabulary (see list below)
 Justify that two triangles are congruent.
 Find congruent triangles, including those in complex diagrams.
 Understand that not all lines cut by a transversal are parallel.
 Reason about angles given parallel lines and about parallel lines given angle
relationships.
 Write proofs using definitions, postulates, and theorems about triangles and parallel lines.
Vocabulary
Acute Angle
Acute Triangle
Adjacent Angles
Altitude
Angle Bisector
Bisect
Trisect
Conjecture
Consecutive Angles
Diagonal
Equiangular
Equilateral
Equilateral Triangle
Exterior Angle of a Polygon
Interior Angle of a Polygon
Invariant
Isosceles Triangle
Median
Midpoint
Obtuse Angle
Obtuse Triangle
Parallel
Perpendicular
Perpendicular Bisector
Right Angle
Right Triangle
Scalene Triangle
Straight Angle
Straight Angle
Supplementary Angles
Complementary Angles
Vertical Angles
Theorems and Postulates for Using in Proofs
Postulates
Linear Pair Postulate
If two angles form a linear pair, then they are supplementary.
mÐ1 + mÐ2 = 180°
2
1
Parallel postulate.
Given a line and a point not on the line there is exactly one line passing
through the point that is parallel to the line.
SAS postulate.
Given two triangles, if two corresponding sides and the corresponding
angle between them are congruent, then the triangles are congruent.
SSS postulate.
Given two triangles, if the three corresponding sides are congruent,
then the triangles are congruent.
ASA postulate.
Given two triangles, if two corresponding angles and the corresponding
side between them are congruent, then the triangles are congruent.
!
!
!
Theorems
HL Congruence,
Given two right triangles, if the corresponding hypotenuse
and leg are congruent, then the triangles are congruent.
AAS Congruence.
Given two triangles, if two corresponding angles and a
corresponding side not between them are congruent, then the
triangles are congruent.
Vertical Angles Theorem.
STAY TUNED …
(not on test)
!
Vertical angles are equal.
Triangle Exterior Angle Theorem.
3
The measure of an exterior angle of a triangle is equal to the
sum of the remote interior angles.
Isosceles Triangle Theorem and its Converse
If a triangle has two congruent sides, then the angles opposite
those sides are also congruent.
If a triangle has two congruent angles, then the sides opposite
those angles are also congruent.
Uniqueness of perpendiculars.
Given a line and a point not on a line there is exactly one line
passing through the point that is perpendicular to the line.
PAI Theorem.
If lines are Parallel, then Alternate Interior angles are
congruent.
Alternate Exterior Angle Theorem.
If lines are Parallel, then Alternate Exterior angles are
congruent.
Corresponding Angle Theorem.
2
mÐ1 = mÐ2 + mÐ3
1
(Use only one arrow in your
proof to specify direction.)
®
®
®
If lines are Parallel, then Corresponding Angles are congruent.
AIP Theorem
If Alternate Interior angles are congruent, then lines are
Parallel.
Triangle Angle Sum Theorem.
The sum of the angles in a triangle is 180
All right angles are congruent
Complements of congruent angles are congruent
Supplements of congruent angles are congruent
2
Definitions and Other Reasons Frequently Used in Proof
Reflexive Property
Any segment is congruent to itself. Any angle is congruent to itself.
Midpoint or Segment Bisector
A midpoint or segment bisector divides a segment into two congruent segments.
Angle Bisector
An angle bisector divides a segment into two congruent angles.
Perpendicular (Lines, segments, rays)
If two lines, segments, or rays are perpendicular, then they intersect to form four right angles.
Definition of Congruent Triangles (CPCTC)
Corresponding Parts of Congruent Triangles are Congruent, or if two triangles are congruent,
then their corresponding parts are also congruent.
This list is necessarily incomplete as we will continue to add to it.
Proof Writing Tips
One of the best ways of figuring out how to write a proof is to work backwards from the
conclusion.
Ask yourself the following questions:
1) What am I trying to prove?
Congruent triangles? Congruent segments? Congruent angles?
Another relationship? (Isosceles triangles, midpoints, bisectors, etc.)
2) What are the possible methods of proving that?
Congruent triangles - SSS, SAS, ASA, AAS
Congruent segments - CPCTC
Congruent angles - CPCTC
Another relationship - fulfill the needs of a definition or a theorem
3) Is this method likely to work here?
Are there triangles with the parts you want? Which ones?
4) Do I have enough information to prove two triangles are congruent?
Label the figure with known information (given as well as inferred).
Do the triangles share a side or angle so that it is congruent in both triangles?
5) Did I use all of the given information?
Does the information given apply more to another pair of triangles?
6) Does every step serve a purpose?
Do you have steps that don’t lead anywhere (except the conclusion)?
7) Does each step have a valid reason? Does it lead to another step correctly?
3
Directions: Mark the given information on the figure. Then complete the proof.
Write your proofs on a separate piece of paper so you have room to be complete!
Hints:
 Mark the diagram with the given information, redraw overlapping triangles separately to help
you see things clearly
 See if the sides or angles you are trying to prove congruent are parts of triangles you can
prove are congruent
1.
̅̅̅̅
Given: ̅̅̅̅
𝑋𝑅 //𝑆𝑌
̅̅̅̅ bisects 𝑋𝑌
̅̅̅̅
𝑅𝑆
R
Prove: Δ𝑅𝑀𝑋 ≅ Δ𝑆𝑀𝑌
Y
M
X
S
D
C
2
̅̅̅̅
2. Given: ̅̅̅̅
𝐷𝐶 //𝐴𝐵
̅̅̅̅//𝐵𝐶
̅̅̅̅
𝐴𝐷
Prove: Δ𝐴𝐵𝐶 ≅ ∆𝐶𝐷𝐴
1
A
3
4
B
3.
4
4.
5.
Given: AB @ CD;
AD @ BC
A
AX ^ BD; CY ^ BD
D
Y
Prove: AX @ CY
X
C
B
6.
Given: BED, Ð1@ Ð2, Ð3 @ Ð4
Prove: BA @ BC
7. Draw a figure and write the given(s) and what you need to prove, then write the proof. Use
proper notation.
The altitude to the base of an isosceles triangle is also a median.
Given:
Labeled Diagram.
Prove:
5
8. Given: MHF , SHK , ME @ SE , MH @ SH
Prove: KH @ FH
9. Given: KT and CD bisect each other
D
K
A
AK @ TB
Prove: AD @ BC
M
C
B
T
10. A student built this GeoGebra diagram
where AD//CE , AB bisects ÐDAC , and
CB bisects ÐECA. She found that when
she moved the points around, the
measure of ÐB was invariant.
What is the measure of ÐB?
Prove your answer.
D
A
B
C
E
Hints: #6) how can the small triangles
help you with your proof? #8 & 9) there
are 3 pairs of triangles, which can you
prove congruent right away? How is that
helpful? #10) think about pairs of
supplementary angles.
6