1. I II X X Statements Reasons 1. 𝐴𝐵 ≅ 𝐴𝐶 1. Given 2. 𝑅𝐵 ≅ 𝑅𝐶 2. Given
1. 𝐴𝐵 ≅ 𝐴𝐶
1. Given II
2. 𝑅𝐵 ≅ 𝑅𝐶
2. Given X X
3. 𝐴𝑅 ≅ 𝐴𝑅
3. ∡𝑍𝑋𝐽≅∡𝑍𝑌𝐾
3. Reflexive Postulate
4. ∆𝐴𝐵𝑅≅∆𝐴𝐶𝑅
4. 𝑆𝑆𝑆≅𝑆𝑆𝑆
5. Corresponding parts of congruent triangles are congruent
5. ∡𝐴𝑅𝐵≅∡𝐴𝑅𝐶
6. Angles on a line
6. ∡𝐵𝑅𝑆 𝑎𝑛𝑑 ∡𝐴𝑅𝐵 are supplements
∡𝐶𝑅𝑆 𝑎𝑛𝑑 ∡𝐴𝑅𝐶 are supplements
7. ∡𝑃≅∡𝑃
7. Complements of congruent angles are congruent
7. ∡𝐵𝑅𝑆≅∡𝐶𝑅𝑆
8. 𝑅𝑆 ≅ 𝑅𝑆
8. Reflexive Postulate
9. ∆𝑅𝐵𝑆≅∆𝑅𝐶𝑆
9. 𝑆𝐴𝑆≅𝑆𝐴𝑆
10. Corresponding parts of congruent triangles are congruent
𝑆𝐵 ≅ 𝑆𝐶

2. I II Statements Reasons 1. 𝑱𝑲 ≅ 𝑱𝑳 1. Given 2. 𝑱𝑿 ≅ 𝑱𝒀 2. Given I
1. 𝑱𝑲 ≅ 𝑱𝑳
1. Given
2. 𝑱𝑿 ≅ 𝑱𝒀
2. Given I
3. ∡𝑲≅∡𝑳
3. In a triangle (KJL) angles opposite congruent sides are congruent
4. ∡𝑱𝑿𝒀≅∡𝑱𝒀𝑿
4. In a triangle (JXY) angles opposite congruent sides are congruent
6. Angles on a line
6. ∡𝑱𝑿𝒀 𝒂𝒏𝒅 ∡𝑱𝑿𝑲 are supplements
∡𝑱𝒀𝑿 𝒂𝒏𝒅 ∡𝑱𝒀𝑳 are supplements
7. ∡𝑱𝑿𝑲≅∡𝑱𝒀𝑳
7. Complements of congruent angles are congruent
8. ∆𝑱𝑲𝑿≅∆𝑱𝑳𝒀
8. 𝑨𝑨𝑺≅𝑨𝑨𝑺
9. 𝑲𝑿 ≅ 𝑳𝒀
9. Corresponding parts of congruent triangles are congruent

3. Lesson 8 - Properties of Parallelograms
Unit 3
Lesson 8 - Properties of Parallelograms

4. No, not based on the immediate evidence.
Is 𝐴𝑇 congruent to 𝐽𝑀 ?
No, not based on the immediate evidence.
What relationship does 𝐴𝑇 have to 𝐽𝑀 ?
𝐴𝑇 ∥ 𝐽𝑀
Is ∡𝑇 congruent to ∡J, why?
Yes, they are alternate interior angles.
∆𝐴𝐺𝑇≅∆𝑀𝑌𝐽
S𝐴𝐴≅𝑆𝐴𝐴 𝐽𝑌

5. - a quadrilateral with both pairs of opposite sides parallel quad
Prior knowledge:
- prefix meaning four
*** four-sided figure
- meaning side
- a quadrilateral with both pairs of opposite sides parallel
quad
lateral

6. Note: A and C can be the obtuse angles or the acute angles1 2 4 3 A B C D
𝐴𝐵𝐶𝐷 𝑖𝑠 𝑎 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙𝑜𝑔𝑟𝑎𝑚 X
Opposite sides and opposite angles are congruent
Statements
Reasons
1. 𝐴𝐵𝐶𝐷 𝑖𝑠 𝑎 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙𝑜𝑔𝑟𝑎𝑚
1. Given D
2. 𝐴𝐵 ∥ 𝐷𝐶 and 𝐴𝐷 ∥ 𝐵𝐶
2. Opposite sides of a parallelogram are parallel
3. ∡1 ≅ ∡2 and ∡3 ≅ ∡4
3. When parallel lines are cut by a transversal the alternate interior angles are congruent
4. 𝐷𝐵 ≅ 𝐷𝐵
4. Reflexive postulate
5. ∆𝐴𝐵𝐷≅∆𝐶𝐷𝐵
5. 𝐴𝑆𝐴≅𝐴𝑆𝐴
6. CPCTC
6. 𝐴𝐷 ≅ 𝐶𝐵 and 𝐴𝐵 ≅ 𝐶𝐷
7. ∡𝐴 ≅ ∡C
7. CPCTC
Therefore, in a parallelogram the opposite sides and opposite angles are congruent

7. l V 𝐴𝐵𝐶𝐷 𝑖𝑠 𝑎 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙𝑜𝑔𝑟𝑎𝑚 The diagonals bisect each other Statements
Reasons
1. ABCD is a parallelogram
1. Given
2. AB ∥ DC
2. Opposite sides of a parallelogram are parallel
3. ∡1 ≅ ∡2 and ∡3 ≅ ∡4
3. When parallel lines are cut by a transversal the alternate interior angles are congruent
4. AB ≅ DC
4. Opposite sides of a parallelogram are congruent
5. ∆ABE≅∆CDE
5. ASA≅ASA
6. CPCTC
6. BE ≅ DE and AE ≅ CE
7. A bisector divides a segment into two congruent segments
7. The diagonals bisect each other

8. II I X Statements Reasons
Quadrilateral ABCD, with opposite sides congruent X
𝑨𝑩𝑪𝑫 𝒊𝒔 𝒂 𝒑𝒂𝒓𝒂𝒍𝒍𝒆𝒍𝒐𝒈𝒓𝒂𝒎 I
Statements
Reasons
1. Quadrilateral ABCD, with opposite sides congruent
1. Given
2. 𝑫𝑩 ≅ 𝑫𝑩
2. Reflexive postulate
3. ∆𝑨𝑩𝑫≅∆𝑪𝑫𝑩
3. 𝑺𝑺𝑺≅𝑺𝑺𝑺
4. ∡𝟏 ≅ ∡2 and ∡𝟑 ≅ ∡4
4. CPCTC
5. When two lines are cut by a transversal, making the alternate interior angles are congruent, the lines are parallel.
5. 𝑨𝑩 ∥ 𝑫𝑪 and 𝑨𝑫 ∥ 𝑩𝑪
6. 𝑨𝑩𝑪𝑫 𝒊𝒔 𝒂 𝒑𝒂𝒓𝒂𝒍𝒍𝒆𝒍𝒐𝒈𝒓𝒂𝒎
6. A quadrilateral with two pairs of opposite sides parallel is a parallelogram

9. Homework – Page 38