5.1 Indirect Proofs
5.2 Proving lines parallel/5.3 Proving angles congruent
5.4 Four Sided Polygons
5.5 Properties of Quadrilaterals
5.6/5.7 Proving figures to be special quadrilaterals
100
Summarize the procedure to completing an indirect proof.
1. List possibilities for conclusion 2. Assume negation of conclusion is correct 3. Write a chain of reasons until you reach a contradiction of the given information or a theorem or known fact. 4. State the remaining possibility as the desired conclusion
100
For each one tell which theroem would be used to prove lines parallel (pg. 219 #1)
a. Corresponding ∠s ≅ b. Alternate interior ∠s ≅ c. Alternate exterior ∠s ≅
100
Find the area of a square whose perimeter is 65 feet
264 units^2
100
Given: parallelogramABCD
Conclusion: ⟁ABC ≅ ⟁CDA
Pg. 244 #1
Statement Reason
1. Given 1. Given
2.∠ABC≅∠CDA 2. Oppo. ∠s of a parallelogram are ≅
3. AB ≅ CD, AD ≅ CB 3. Oppo sides of a parallelogram are ≅
4. ⟁ABC ≅ ⟁CDA 4. SAS
100
The measure of one angle of a parallelogram is 40 more than 3 times another.
Find the measure of each angle
35, 145
200
Given: AB ≅ AD, ∠BAC ≆ ∠DAC
Prove: BC ≆ DC
(pg. 213 #1)
Statement Reason
1. Given 1. Given
2. Assume BC ≅ DC 2. Assume opposite of to prove
3. AC ≅ AC 3. Reflexive Property
4. ⟁ABC ≅ ⟁ADC 4. SSS
"5.∠BAC ≅ ∠DAC" 5.CPCTC CONTRADICTION
6. BC ≆ DC 6. PIR
200
Write an inequality and find the restrictions on x (hint: use remote interior angles) pg. 222 #17
30<3x-18<180 48<3x<198 16<x<66
200
Given: ABCD is a kite pg. 239 #16 AB = x + 3 BC = x + 4 CD = 2x-1 AD= 3x-y What is the perimeter of the kite?
7
200
Given: parallelogram WSTV WS= x+5 WV= x+9 VT= 2x+1 Find the perimeter of WSTV
P=44 units
200
Given: RSOT is a parallelogram, MS ≅ TP
Prove: MOPR is a parallelogram, pg. 253. #14
Statement Reason
1. Given 1. Given
2. RS ≅ TO 2. Oppo sides of a parallelogram are ≅
3. RS II TO 3. Oppo sides of a parallelogram are II
4. ∠RSM ≅∠PTO 4. II lines → AeA ≅
5. ⟁RSM ≅ ⟁OTP 5.SAS
6. RM ≅ PO 6. CPCTC
7. SO ≅ RT 7. Oppo sides of a parallelogram are ≅
8. RT II SO 8. Oppo sides of a parallelogram are II
9. ∠MSO ≅ ∠RTP 9. II lines → AeA ≅
10. ⟁RTP ≅ ⟁OSM 10.SAS
11. MO ≅ RP 11. CPCTC
12. MOPR is a parallelogram 12. Oppo sides ≅→ quad is a parallelogram
300
Given: PA ⊥ AB, PA ⊥ AC, ∠B ≅ ∠C
Prove: AB ≆ AC (Pg. 214 #10)
Statement Reason
1. Given 1. Given
2. Assume AB ≅ AC 2. Assume opposite of to prove
3. ∠PAB ≅ ∠PAC 3. ⊥ lines form ≅ Right
4. PA ≅ PA 4. Reflexive Prop
5. ⟁PAB ≅ ⟁PAC 5.SAS
6. ∠B ≅ ∠C 6. CPCTC
CONTRADICTION
7.AB ≇ AC 7.PIR
300
(A crook problem) If f ‖ g, find m∠8 pg. 231 #10
m∠8= 150º
300
Say whether it is always (A), sometimes (S), never (N) a. A square is a rhombus b. A rhombus is a square c. A kite is a parallelogram d. A rectangle is a polygon
a. A b. S c. S d. A
300
Given: VRST is an isos. trapezoid with legs VR and TS
Prove: ⟁ARS is isos.
Statement Reason
1. Given 1. Given
2. VR≅TS 2. legs of an isos trapezoid are ≅
3. VS ≅ TR 3. Diagonals of an isos trapezoid are ≅
4. RS≅RS 4. Reflexive Prop
5. ⟁VRS≅⟁TRS 5.SSS
6.∠VSR ≅ ∠TRS 6. CPCTC
7. AR≅AS 7. If angles then sides
8. ⟁ARS is isos. 8. Def of isos.
300
Give the most descriptive name for:
a. A quadrilateral with diagnols that are perpendicular bisectors of each other
b. A rectangle that is also a kite
a. rhombus b. square
400
Given: ⊙O; HE is not the ⊥ bis of DF
Prove: DE ≇ EF (pg. 214 #12)
Statement Reason
1. Given 1. Given
2. Assume DE ≅ EF 2. Assume opposite of to prove
Draw OD and OF 3. Two Points determine a line
4. OD ≅ OF 4. All Radii of a circle are ≅
5. HE ⊥ bis DF 5. Equidistance Theorem
CONTRADICTION
6. DE ≇ EF 7.PIR
400
Given: FH ‖ JM, ∠1 ≅ ∠2, FH ≅ JM
Prove: GJ ≅ HK (pg. 231 #15)
Statement Reason
1. Given 1. Given
2. ∠HFG ≅ ∠KMJ 2. Supplements of ≅ ∠s are ≅
3. ∠FHG ≅ ∠KJM 3. II lines→AIA ≅
4. ⟁FHG ≅ ⟁MJK 4. ASA
5. GH ≅ JK 5. CPCTC
6. HJ ≅ HJ 6. Reflexive Prop
7. GJ ≅ HK 7. Addition Prop
400
State whether it is Always (A), Sometimes (S), or Never (N) a. A polygon has the same number of sides as vertices b. A parallelogram has three diagonals c. A trapezoid has three bases
a. A b. N c. N
400
Given: RHOM is a rhombus
m∠MBR= 21x+13, MR= 6.2x
Find the length of RH to the nearest tenth
22.7
400
Given: ID bis. RB, BI ≅ IR
Prove: BIRD is a kite, pg. 259 #10
Statement Reason
1. Given 1. Given
2. BK ≅ KR 2. Def of segment bisection
3. ID ⊥ bis. BR 3. Equidistance theorem
4. BIRD is a kite 4. If one of the diagnols of a quad is the ⊥ bis. of the other diagonal, then it is a kite
500
Prove that if ⟁ABC is isosceles, with base BC, and if P is a point on BC that is not the midpoint, then ⃗ray AP does not bisect ∠BAC
Statement Reason
1. Given:Isos ⟁ABC with base BC, P is not the midpoint of BC 1. Given
2. Assume AP bis. ∠BAC 2. Assume opposite of to prove
3. AB ≅ AC 3. Def of Isos
4. AP ≅ AP 4. Reflexive Prop
5. ⟁ABP ≅ ⟁ACP 5. SAS
6. BP ≅ PC 6.CPCTC
7. P midst of BC 7. Def of midst
CONTRADICTION
8. ⃗ray AP does not bisect ∠BAC 8.PIR
500
If two parallel lines are cut by a transversal, eight angles are formed (not counting straight angles) a. If two angles are chosen at random, what is the probability that these angles will be congruent?
2/7
500
How many rectangles are shown in the figure at the right, in which all of the angles are right angles? Pg. 240 #20
20 rectangles
500
RHOM is a rhombus
a. Find the coordinates of O
(19, 15)
500
What is the most descriptive name for the quadrilateral with vertices (3,2), (8,1), (7,6), and (2,7)?
Rhombus
M
e
n
u