Major EOC Concepts: Proofs
May 8, 2014
Theorems: You will be asked to fill in missing parts of proofs, put proofs in correct logical order, or to
correct steps in a proof. Remember that proofs always start with the given and end with the prove. Read
carefully and mark all information on the picture to help you fill in the proof. Think about what you
know based on the information given. Continue in this manner until you can reach the “prove”.
HINTS:
-When lines are parallel, look for a transversal to make congruent alternate interior angles.
-Are the triangles in the picture sharing a side or angle? Use the reflexive property.
-To prove any of the parallelogram theorems, first prove the triangles congruent, then use
CPCTC to prove the theorem.
1.) The following is a list of general theorems that you might need.
*Transitive Property: If a = b and b = c, a = c. If two things are equal to the same thing, they are
equal.
*Reflexive property: when an object is congruent to itself.
*Bisect or midpoint: cut into two congruent pieces
*Substitution: substitute one object for another
*CPCTC: Corresponding parts of congruent triangles are congruent. Use AFTER two triangles are
proven congruent and you want to say that their individual parts are congruent.
*Definition of Similarity: Sides of similar triangles are proportional and the angles are congruent.
Use AFTER two triangles are proven similar to prove proportional sides or congruent angles.
2.) Theorems about lines and angles.
Perpendicular: two lines that intersect at a right angle
Vertical angles: When two lines cross, the angles across from each other are congruent.
*Parallel lines and tranversals: When a transversal crosses parallel lines, alternate interior and
corresponding angles are congruent.
*Perpendicular bisector: Points on a perpendicular bisector of a line segment are equidistant
from the segment’s endpoints; right angles are created where a perpendicular bisector and a
line segment meet.
2.) Theorems about triangles.
Triangle sum: The measures of interior angles of a triangle sum to 180°
Isosceles triangles: Base angles of isosceles triangles are congruent and the sides across from
the base angles are congruent.
*Congruency Theorems: Side Angle Side, Side Side Side, Angle Side Angle, and Angle Angle Side.
Congruent figures have congruent angles and sides.
3.) Prove theorems about parallelograms. This shape family includes parallelograms, rectangles, rhombi
(rhombus) and squares. Opposite sides are congruent and parallel, opposite angles are congruent, the
diagonals of bisect each other, and rectangles and squares have congruent diagonals.
Name________________________
Question of the Day: Proofs
May 8. 2014
Match the definition with a term or theorem from the box. Place the letter in the blank
1.) _______: divides an angle into two congruent
adjacent angles
13.) ______: corresponding sides of two triangles
are congruent
2.) _______: equidistant from the endpoints of a
segment, gives two equal peices of the segment
14.) ______: divide into two congruent peices
3.) _______: something is congruent to itself
15.) ______: Angles on the same location of the
parallel lines
4.) _______: If two things are congruent to the
same thing, then they are congruent.
16.) ______: pieces of congruent triangles are
congruent, used after two triangles are proven
congruent
5.) _______: intersects a segment at its midpoint
and forms right angles
17.) _______: two angles that add to be 90
6.) _______: angles inside the parallel lines on
opposite sides of the transversal
7.) _______: two pairs of corresponding sides and
the angle between them are congruent
18.) _______: two adjacent angles that form a line
19.) _______: add up pieces of a segment to get
the entire segment
20.) _______: the angles of a triangle add to 180
8.) _______: angles opposite each other when two
lines cross
14.) ______: add up the small pieces of an angle to
get the whole angle
9.) _______: In two triangles, two pairs of
corresponding angles and the side next to one of
the corresponding angles are congruent
21.) ______: two angles that add to be 180
10.) ______: lines that intersect at right angles
11.) ______: In two triangles, two pairs of
corresponding angles and the side between them
are congruent
12.) ______: In two triangles, two congruent sides
and its base angles are congruent
22.) ______: two or more lines that never
intersect
23.) _____: If two expressions are equal, one may
be used for the other.
24.) _____: If two triangles are similar, their sides
are proportional and angles are congruent.
a. reflexive property
b. angles on a line
c. midpoint
d. transitive property
e. vertical angles
f. complementary angles
g. angle addition
h. angle angle side
i. angle side angle
j. segment addition
k. perpendicular bisector
l. angles at a point
m. supplementary angles
n. side side side
o. alternate interior
p. alternate exterior
q. triangle sum
r. side angle side
s.equilateral triangle
t. isosceles triangle
u. perpendicular lines
v. parallel lines
w. substitution property
x. angle bisector
y. CPCTC
z. definition of similarity
Major Cluster Example Problems
Proofs
1.) Tina is trying to prove the theorem that opposite sides of a parallelogram are congruent.
Given: PQRS is a parallelogram.
Prove: PQ ≅ RS, QR ≅ SP
Tina’s unfinished proof is shown. Finish her proof:
2.) Jorge is proving the theorem that states that all the points on a perpendicular bisector of a line
segment are equidistant from the segment’s endpoints.
Given:
is the perpendicular bisector of
.
Prove:
A.) What statement and reason should go at A?
B.) What statement and reason should go at B?
3.) Complete the incomplete proof:
D
A
Given: X is the midpoint of BD
X is the midpoint of AC
X
Prove: < 𝐷𝑋𝐴 ≅< 𝐵𝑋𝐶
C
B
Statements
Reasons
1. X is the midpoint of BD ; X is the
midpoint of AC .
1.
2.
2.
3.
3. Vertical Angles
4.
4.
5. <DXA ≅ <BXC
5.
4.) Finish the proof:
5.)Correct any mistakes in the proof given
Given: LO ll PM, LP bisects OM.
Prove: 𝐿𝑂 ≅ 𝑃𝑀
Statements
1.) LO ll PM, LP bisects OM.
Definition of parallel
Reasons
2.) LN ≅ PN
Definition of midpoint
3.) <OLN ≅ <PMN
Alternate interior angles
4.) <LNO ≅ <PNM
Vertical Angles
5.) ∆𝑂𝑁𝐿 ≅ ∆𝑃𝑁𝑀
SAS
6.) 𝐿𝑂 ≅ 𝑃𝑀
Definition of Congruence
-What other theorem could be used to prove congruence? What information would be changed?
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6.)
Given: Triangle ABC on line segment AD
Prove: m∠A + m∠B = m∠BCD
Statement
Reason
1.
1.
2.
2.
3.
m∠BCA + m∠BCD = 180
Angle sum of a triangle
3.
4.
4.
5.
5.
Transitive Property
7.)
Given: BA is the perpendicular bisector of RT
Prove: ̅̅̅̅
𝑅𝐴 ≅ ̅̅̅̅
𝑇𝐴
Statements
1.)
Given
1.)
2.)m<ABT=90; m<ABR=90
2.)
3.)
3.)Transitive Property
4.) < 𝐴𝐵𝑇 ≅< 𝐴𝐵𝑅
4.)
5.) 𝐵𝐴 ≅ 𝐵𝐴
5.)
6.)
6.)Def of perpndicular bisector
7.) ∆RBA ≅ ∆TBA
7.)
8.)
8.)
8.) Complete the proof given:
Statements
1.)
2.) < 𝐷𝐸𝐶 ≅ < 𝐵𝐴𝐶
3.) < 𝐷𝐶𝐸 ≅ < 𝐵𝐶𝐴
4.)
𝐷𝐶
5.) 𝐵𝐶 =
𝐷𝐸
𝐵𝐴
Reasons
9.) If a quadrilateral is a parallelogram, the 2 pairs of opposite sides are congruent.
Statements
1 Parallelogram ABCD
Given
Reasons
2
2. Def of parallelogram
3
3 alternate interior angles
4 AC ≅ AC
4
5
5
6
6
10.) Complete the two column proof:
̅̅̅̅, then PQ =
If Q is the midpoint of 𝑃𝑅
1
2
𝑃𝑅.
P
Q
Statements
R
Reasons
Definition of midpoint
Segment addition
PQ + PQ = PR
Multiplication