What was your goal? What was your actual grade? Why did you meet/not meet your goal? What were your strengths? What areas do you need to work on? What are you going to do to succeed

on the next test?

Angles of a Triangle

Write down everything you remember about triangles!

By Side:› Equilateral—all sides congruent› Isosceles—two sides congruent› Scalene—no sides congruent

By Angle:› Obtuse—one angle greater than 90°› Right—one angle equal to 90°› Acute—all angles less than 90°

Interior Angles add up to 180° Exterior Angles add up to 360° Examples:

› In triangle DEF; ∠D = 45°, ∠E = 30°. Find ∠F.

› In triangle ABC; ∠C = 3x – 5, ∠B = x + 40 and ∠A = 2x + 25. Find the measures of all three angles. 2x + 5

3x – 5 x + 40

CPCTC and SSS

Congruent line segments are marked with a small dash

Congruent angles are marked with an arc

Parallel lines are marked with arrows

To separate different pairs of congruent line segments or angles, we use different numbers of dashes or arcs

Corresponding = matching Order of the letters matters! Example: ΔABC = ΔDEF. Which angles are

congruent? Which sides are congruent?

SSS Postulate: If all three corresponding sides of two triangles are congruent, then the triangles are congruent

Example: ΔFEG = ΔKJL because of SSS.

SAS, ASA, AAS, HL

Name all the corresponding angles if ΔIJH ≅ ΔKJL.

∠SRU ≅ ∠STU ∠RSU ≅ ∠TSU ∠RUS ≅ ∠TUS

Yes, can be proven through SSS

∠BRD ≅ ∠DYB ∠RBD ≅ ∠YDB ∠RDB ≅ ∠YBD

BR ≅ DY BY ≅ DR BD ≅ BD

Side-Side-Side (SSS)

Side-Angle-Side (SAS)› Sandwich!

Angle-Side-Angle (ASA)› Sandwich!

Angle-Angle-Side (AAS)› No sandwich!

Hypotenuse-Leg (HL)› Right triangles only!

Identity Properties in Triangle Proofs

Reflexive Property: AB ≅ AB (congruent to itself)

Transitive Property: AB ≅ BC, BC ≅ CD, so AB ≅ CD

Additive Property: Adding the same amount to two congruent parts results in two equal sums

Multiplicative Property: Multiplying two congruent parts by the same number results in two equal products

1. Mark diagram with “Given” and write as Step 1.2. Figure out how many parts of the triangles you

know are congruent, and how many you need to prove congruent.

3. Mark missing congruent parts on diagram, using info from theorems you know (vertical angles, etc.). Write these down in the two columns.

4. Prove triangles congruent using: SSS, SAS, ASA, AAS, or HL.

5. Check: Make sure you used all info in the “Given.” Make sure your last step matches the “Prove”.

Given: GJ ≅ JIHJ ┴ GI Prove: ΔGJH ≅ ΔIJH

Statement Reason

1. GJ ≅ JIHJ ┴ GI

1. Given

2. ∠GJH ≅ ∠IJH2. ┴ lines form right angles, all right angles are ≅

3. HJ ≅ HJ 3. Reflexive property

4. ΔGJH ≅ ΔIJH 4. SAS

Line/Angle Theorems in Triangle Proofs

Midpoint› Halfway point on a line

segment Bisect

› Split a line segment or angle into two equal parts

HJ bisects GI

V is the midpoint of TW

Vertical Angles › ALWAYS congruent; (“X”)

Alternate Interior Angles› ONLY congruent when we know lines are

parallel (“Z”)› ABCD is a

parallelogram

1. Mark diagram with “Given” and write as Step 1.2. Figure out how many parts of the triangles you

know are congruent, and how many you need to prove congruent.

3. Mark missing congruent parts on diagram, using info from theorems you know (vertical angles, etc.). Write these down in the two columns.

4. Prove triangles congruent using: SSS, SAS, ASA, AAS, or HL.

5. Check: Make sure you used all info in the “Given.” Make sure your last step matches the “Prove”.

Given: HK bisects IL ∠IHJ ≅ ∠JKL. Prove: ΔIHJ ≅ ΔLKJ

Statement Reason

1. HK bisects IL∠IHJ = ∠JKL

1. Given

2. IJ ≅ JL 2. Definition of “bisect”

3. ∠IJH ≅ ∠LJK 3. Vertical angles congruent

4. ΔIHJ ≅ ΔLKJ 4. AAS

Using Quadrilateral Theorems in Triangle Proofs

Parallelogram Rhombus Rectangle Square

ALSO WATCH OUT FOR:› Alternate Interior Angles› Vertical Angles

• Opposite sides are parallel and congruent

• •

• Opposite angles are congruent

• • Diagonals bisect each other

• Bisect = to split in half

CBADCDAB ,

CBADCDAB ,

CDAABCBCDDAB ,

• Has all the properties of a parallelogram, plus:• FOUR congruent

sides• Diagonals are

perpendicular and bisect

• Has all properties of a parallelogram, plus:• Four right angles• Congruent

diagonals that bisect

• Four congruent sides and four right angles

• Diagonals are congruent and perpendicular; also bisect

Given: FLSH is a parallelogram; LG ┴ FS, AH ┴ FS

Prove: ΔLGS ≅ ΔHAFStatement Reason

1. FLSH is a parallelogram; LG ┴ FS, AH ┴ FS

1. Given

2. LG ≅ FH 2. Opp. sides of p.gram are ≅

3. ∠LGS ≅ ∠HAF 3. ┴ lines form right ∠’s, all right ∠’s ≅

4. ∠LSG ≅ ∠HFA 4. Alt. int. ∠’s ≅ when lines ||

5. ΔLGS ≅ ΔHAF 5. AAS

Using Circle Theorems in Triangle Proofs

Chords intercepting congruent arcs are congruent

Tangent is perpendicular to the radius at the point where it touches the circle

Arcs between parallel lines are congruent

H

F

J

G

Inscribed angle is half the intercepted arc.

Two inscribed angles that intercept the same arc are congruent

N

J

LM

Given: arc BR = 70°, arc YD = 70°; BD is the diameter of circle O

Prove: ΔRBD ≅ ΔYDBStatement Reason

1. Arc BR = 70°, arc YD = 70°; BD is the diam. of circle O

1. Given

2. BD ≅ BD 2. Reflexive

3. ∠YBD = 35°, ∠RDB = 35°

3. Inscribed angles = ½ arc

4. ∠YBD ≅ ∠RDB 4. ≅ arcs have same measure

5. ∠BYD ≅ ∠BRD5. Inscribed angles intercepting same arc are ≅

5. ΔRBD ≅ ΔYDB 6. AAS