Unit 4
For help with unit 4.5 - 4.7 See Unit 3
Unit 4 Study Guide
Polygon Congruence Postulate: Two polygons are congruent if and only if there is a correspondence between their sides and angles such that:
Each pair of corresponding angles is congruent.
Each pair of corresponding sides is congruent.
SSS (Side-Side-Side) Postulate
If the sides of one triangle are congruent to the sides of another triangle, then the two triangles are congruent.
SAS (Side-Angle-Side) Postulate
If two sides and their included angle in one triangle are congruent to two sides and their included angle of another triangle, then the two triangles are congruent.
ASA (Angle-Side-Angle) Postulate
If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.
AAS (Angle-Angle-Side) Congruence Theorem
If two angles and a nonincluded side of one triangle are congruent to the corresponding angles and nonincluded side of another triangle, then the triangles are congruent.
HL (Hypotenuse-Leg) Congruence Theorem
If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of another right triangle, then the two right triangles are congruent.
Isosceles Triangle Theorem
If two sides of a triangle are congruent, then the angles opposite those sides are congruent. Go Back
Converse of the Isosceles Triangle Theorem.
If two angles of a triangle are congruent, then the sides opposite those angles are congruent. Triangle DEF
Equilateral triangle Corollary.
The measures of each angle of an equilateral triangle is 60 degrees.
Vertex Angle Bisector Corollary.
The bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base.
The triangle midsegment theorem:
A midsegment of a triangle is parallel to a side of the triangle and has a measure equal to half of the measure of that side.
Using triangle congruence to find unknown measures.
If the measures of <A and <B = 25 degrees, the sum of their measures is 50 degrees. There are 180 degrees in each triangle; therefore the measure of <C is 180 degrees - 50 degrees or 130 degrees.
D
F Given: <E @ <F, and DE = 10. Find the length of DF.
E
XYZ is an equiangular triangle where <X @ <Y @ <Z \m<X = m<Y = m>Z and
XY = YZ= ZX. If YZ = 12, then XY = 12 AND ZS = 12.
KLM is an isosceles triangle where <L @ <M \m<L = 35°, then m<M = 35° and m<K = 180° - (35°(2)). The m<K = 180 - 70 or 110°.
Paragraph proof: PQR is an isosceles triangle where ,m<R = m<S and m<S = 3x; m<P = 6x (given). The measure of the sum of the angles of a triangle is 180 degrees. So, 3x + 3x + 6x = 180. When we combine like terms we find that 12 x = 180. Then by the division property of equality x = 15 degrees. By substituting 15 for x, we learn that the measure of <P is 90 degrees -- 6x = 6(15) or 90 degrees -- and the measure of <Q is 45 degrees -- 3x = 3(15) or 45 degrees.
Complete the flow chart proof
Given: and
Prove:
Proof:
Extra practice.
Prove the following:
1. Given: ABCD (ABCD is a parallelogram)
Prove: ABC @ CDA
Proof:
Statement
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Reason
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1. Parallelogram ABCD
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Given
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2. AD @ CB
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2.
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3. ___ ___
DC @ AB
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3.
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4. Construct line AC
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4. One and only one line can be drawn through any two points.
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5. ___ ___
AC @ AC
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5.
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6. ABC @ CDA
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6.
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(Note: line AC is not given in the hypothesis; therefore it is an auxiliary line drawn to help establish the proof. It is a dotted line because it is constructed rather than “given.”)
2. Given:EFGH is a parallelogram; <1 @ <2
Prove: KH = EG
Proof:
Statement
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Reason
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Given
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Opposite angles of a parallelogram @
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Opposite angles of a parallelogram @
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Given
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ASA
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KH = EG
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Segment Congruence Postulate
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3. Given: Rectangle MPRS; MO @ PO
Prove: Triangle ROS is isosceles.
Proof
Statement
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Reason
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1
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Given
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2
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Opposite sides of a rectangle are congruent.
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3
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Given
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4
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The anglesof a rectangle are right angles
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5
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Right angles are congruent.
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6
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SAS
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7
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CPCTC
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8
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If two sides of a triangle are congruent, the triangle is isosceles.
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4. Given: Parallelogram ABCD. Prove: AE @ CF
Proof:
Statement
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Reason
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1. ABCD is a parallelogram
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Given
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2. Segment AD is congruent to segment CB
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3
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Given
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4 Segment AC is a transversal of segments DC and AB
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Definition of a transversal.
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5. Segments DC and AB are parallel
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6
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Alternate Interior Angles are congruent
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7
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SAS
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8. Segment AE @ segment CF
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5. Given: Parallelogram ABCD. Prove: DE ¦ FB
(hint: You will use the converse of the alternate interior angle theorem.) You may not need all the rows provided below.
Proof:
6. Given: Parallelogram WSTV; WS = x + 5, WV = x + 9 VT = 2x + 1.
Find the perimeter of WSTV. Solve it in proof format.
(Hint use the formula P = 2l + 2w where l is the length and w is the width.)
7. Given: Parallelogram ABCD; <a = x°; <D = (3x+4)°.
Find: m<D and m<C.
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