1. GEOMETRY
SOL REVIEW
PACKET
Part 2

2. What I Need to Know - Triangles
Large Large; Small Small
Large side is opposite large angle; small side opposite small angle
Large Large; Small Small
Large angle is opposite large side; small angle opposite small side
Small + Small > Large
The two small sides must have a total length greater than large side.

3. What I Need to Know - Triangles
42 – 20 = 22
= 62
22<𝑥<62
x MUST be greater than 22 AND less than 62.
REMEMBER!! 22 and 62 are NOT on the list of possible lengths!!!

4. What I Need to Know - Triangles9
600 4
800
The only possible combination of sides is:
4, 7, and 9.
These are the only measurements which make small + small > large.
Since angles add up to 180, the other two angles have to be 60 and 80.
Since 400 is the smallest angle, the side opposite it must be the shortest side (4).
The 600 angle has to be opposite the 7, since that will be the middle length.
The 800 angle is opposite the 9, the longest side.

5. What I Need to Know - Triangles4 22
13−9<𝑥<13+9
4<𝑥<22

6. What I Need to Know - Triangles
𝑱𝑲 , 𝑺𝑱 , 𝑺𝑲
2𝑦+6+8𝑦+10+𝑦+32=180
11𝑦+48=180
11𝑦=132
𝑦=12
Next, we plug 12 into each angle to find its actual measurement.
To find the segments, we need to know what y is.
We know that the angles of a triangle add up to 1800, so set them equal to 180.
300
1060
440 =𝑆
30=𝑆
12+32=𝐾
44=𝐾
8(12)+10=𝐽
106=𝐾
The smallest angle is angle S; the shortest side will be JK, the segment opposite angle S.
The middle angle is angle K; the middle side will be SJ, the segment opposite angle K.
The biggest angle is angle J; the longest side will be SK, the segment opposite angle J.

7. What I Need to Know – Congruent Triangles
SSS
ASS
AAA
SAS
ASA
Reflexive Property, Midpoint of a Segment, Symmetric Property, Transitive Property, Alternate Interior Angles, Corresponding Angles, Base Angles of Isosceles Triangle, Segment Bisector, Angle Bisector, Substitution Property
AAS HL
Distance Formula

8. What I Need to Know – Congruent Triangles
Corresponding Parts of Congruent Triangles are Congruent

9. What I Need to Know – Congruent Triangles

10. What I Need to Know – Congruent Triangles
We have to look at the order of the letters in the congruence statement to identify corresponding sides.
Side ST corresponds to side XY
Side TW corresponds to side YZ
Side WS corresponds to side ZX
Set either of these corresponding sides equal to each other and solve for x.
3𝑥−1=4𝑥−5
−1=𝑥−5
4=𝑥
2𝑥+1=𝑥+5
𝑥+1=5
𝑥=4
3𝑥+1=4𝑥−3
1=𝑥−3
4=𝑥

11. What I Need to Know – Congruent Triangles
We are looking the point that corresponds with point C.
Since point C is 4 units away from point B, point F will be 4 units away from E.
Four units away from point E is point (-4, -8).
Connect the points to get a triangle that is congruent to ABC.

12. What I Need to Know – Congruent Triangles
We are looking the point that corresponds with point C.
Since point C is 4 units away from point B, point F will be 4 units away from E.
Four units away from point E is point (-8, -4).
Connect the points to get a triangle that is congruent to ABC.
TWO ANSWERS!!
(-4,-8) and/or (-8,-4)

13. What I Need to Know – Congruent Triangles
Statements
1. 𝐴𝐵 ∥ 𝐶𝐷 , 𝐴𝐹 ≅ 𝐹𝐷
2. ∠𝐵𝐴𝐹≅∠𝐶𝐷𝐹
3. ∠𝐴𝐹𝐵≅∠𝐷𝐹𝐶
4. △𝐴𝐵𝐹≅ΔDCF
Reasons
1. Given
2. If parallel lines are cut by a transversal, then pairs of alternate interior angles are congruent.
3. Vertical angles are ≅.
4. Angle-Side-Angle (ASA) Postulate
(other solutions are possible)

14. What I Need to Know – Similar Triangles
Angle-Angle AA
Side-Side-Side
SSS
Side-Angle-Side
SAS

15. What I Need to Know – Similar Triangles
Corresponding sides are proportional means that if you find the scale factor of the corresponding longer sides, the scale factor of the corresponding smaller sides, and the scale factor of the corresponding medium sides, they should all be equivalent. Corresponding angles are congruent means that the angles in the same position in both triangles should have the same measure.
Scale factor = ratio of corresponding sides
Length of side of △1
Corresponding length of side of △2
𝑺𝒎𝒂𝒍𝒍 𝒔𝒊𝒅𝒆 𝒐𝒇 ∆𝟏 𝑺𝒎𝒂𝒍𝒍 𝒔𝒊𝒅𝒆 𝒐𝒇 ∆𝟐 = 𝑩𝒊𝒈 𝒔𝒊𝒅𝒆 𝒐𝒇 ∆𝟏 𝑩𝒊𝒈 𝒔𝒊𝒅𝒆 𝒐𝒇 ∆𝟐

16. What I Need to Know – Similar Triangles
We have to look at the order of the letters in the similarity statement to identify corresponding sides.
Side AB corresponds to side DE
Side BC corresponds to side EF
Side AC corresponds to side DF
Put the lengths of AB and DE together, since they correspond.
Do the same for the lengths of BC and EF.
Since side EF is half the length of side BC, we know that side DE will be half of side AB.
Side AB is 36, so side EF will be 18.

17. What I Need to Know – Similar Triangles
These will not always be so easy, so you may have to set up a proportion and solve for x.
36 𝑥 = 30 15
30𝑥=540
𝑥=18
This is the same answer we got a minute ago; we just found it a different way.

18. What I Need to Know – Similar Triangles
∠𝑪𝑫𝑨≅∠𝑩𝑨𝑫;
∠𝑪𝑩𝑨≅∠𝑩𝑪𝑫
Angle-Angle (AA)
Postulate

19. What I Need to Know – Similar Triangles
We know that angles AFB and DFC are congruent (vertical angles).
We know that angles FAB and FDC are congruent (alt. interior angles).
We know that angles ABF and DCF are congruent (alt. interior angles).
We have two similar triangles based on AA (Angle-Angle)
Notice that while we have all three angles congruent, we only needed two to prove similarity.
This is because mathematically, if two angles are congruent, the third must also be congruent (they don’t add up 180 if it’s not).