Geometry Section 4-4 1112

Section 4-4

Proving Congruence: SSS, SAS

Monday, February 6, 2012

Essential Questions

How do you use the SSS Postulate to test for triangle congruence?

How do you use the SAS Postulate to test for triangle congruence?

Monday, February 6, 2012

Vocabulary1. Included Angle:

Postulate 4.1 - Side-Side-Side (SSS) Congruence:

Postulate 4.2 - Side-Angle-Side (SAS) Congruence:

Monday, February 6, 2012

Vocabulary1. Included Angle: An angle formed by two adjacent sides of a

polygon

Postulate 4.1 - Side-Side-Side (SSS) Congruence:

Postulate 4.2 - Side-Angle-Side (SAS) Congruence:

Monday, February 6, 2012

Vocabulary1. Included Angle: An angle formed by two adjacent sides of a

polygon

Postulate 4.1 - Side-Side-Side (SSS) Congruence: If three sides of one triangle are congruent to three corresponding sides of a second, then the triangles are congruent

Postulate 4.2 - Side-Angle-Side (SAS) Congruence:

Monday, February 6, 2012

Vocabulary1. Included Angle: An angle formed by two adjacent sides of a

polygon

Postulate 4.1 - Side-Side-Side (SSS) Congruence: If three sides of one triangle are congruent to three corresponding sides of a second, then the triangles are congruent

Postulate 4.2 - Side-Angle-Side (SAS) Congruence: If two sides and the included angle of one triangle are congruent to two corresponding sides and included angle of a second triangle, then the triangles are congruent

Monday, February 6, 2012

Example 1Prove the following.

Given: QU ≅ AD, QD ≅ AU

Prove: QUD ≅ADU

Monday, February 6, 2012

Example 1Prove the following.

Given: QU ≅ AD, QD ≅ AU

Prove: QUD ≅ADU

1. QU ≅ AD, QD ≅ AU

Monday, February 6, 2012

Example 1Prove the following.

Given: QU ≅ AD, QD ≅ AU

Prove: QUD ≅ADU

1. QU ≅ AD, QD ≅ AU 1. Given

Monday, February 6, 2012

Example 1Prove the following.

Given: QU ≅ AD, QD ≅ AU

Prove: QUD ≅ADU

1. QU ≅ AD, QD ≅ AU 1. Given

2. DU ≅ DU

Monday, February 6, 2012

Example 1Prove the following.

Given: QU ≅ AD, QD ≅ AU

Prove: QUD ≅ADU

1. QU ≅ AD, QD ≅ AU 1. Given

2. DU ≅ DU 2. Reflexive

Monday, February 6, 2012

Example 1Prove the following.

Given: QU ≅ AD, QD ≅ AU

Prove: QUD ≅ADU

1. QU ≅ AD, QD ≅ AU 1. Given

2. DU ≅ DU 2. Reflexive

3. DU ≅ UD

Monday, February 6, 2012

Example 1Prove the following.

Given: QU ≅ AD, QD ≅ AU

Prove: QUD ≅ADU

1. QU ≅ AD, QD ≅ AU 1. Given

2. DU ≅ DU 2. Reflexive

3. DU ≅ UD 3. Symmetric

Monday, February 6, 2012

Example 1Prove the following.

Given: QU ≅ AD, QD ≅ AU

Prove: QUD ≅ADU

1. QU ≅ AD, QD ≅ AU 1. Given

2. DU ≅ DU 2. Reflexive

3. DU ≅ UD 3. Symmetric

4. QUD ≅ADU

Monday, February 6, 2012

Example 1Prove the following.

Given: QU ≅ AD, QD ≅ AU

Prove: QUD ≅ADU

1. QU ≅ AD, QD ≅ AU 1. Given

2. DU ≅ DU 2. Reflexive

3. DU ≅ UD 3. Symmetric

4. QUD ≅ADU 4. SSS

Monday, February 6, 2012

Example 2∆DVW has vertices D(−5, −1), V(−1, −2), and W(−7, −4). ∆LPM

has vertices L(1, −5), P(2, −1), and M(4, −7). Graph both triangles on the coordinate plane. Then, using your drawing,

determine whether the triangles are congruent or not, providing a logical argument for your statement.

Monday, February 6, 2012

Example 2∆DVW has vertices D(−5, −1), V(−1, −2), and W(−7, −4). ∆LPM

has vertices L(1, −5), P(2, −1), and M(4, −7). Graph both triangles on the coordinate plane. Then, using your drawing,

determine whether the triangles are congruent or not, providing a logical argument for your statement.

x

yMonday, February 6, 2012

Example 2∆DVW has vertices D(−5, −1), V(−1, −2), and W(−7, −4). ∆LPM

has vertices L(1, −5), P(2, −1), and M(4, −7). Graph both triangles on the coordinate plane. Then, using your drawing,

determine whether the triangles are congruent or not, providing a logical argument for your statement.

x

y

D

Monday, February 6, 2012

Example 2∆DVW has vertices D(−5, −1), V(−1, −2), and W(−7, −4). ∆LPM

has vertices L(1, −5), P(2, −1), and M(4, −7). Graph both triangles on the coordinate plane. Then, using your drawing,

determine whether the triangles are congruent or not, providing a logical argument for your statement.

x

y

DV

Monday, February 6, 2012

Example 2∆DVW has vertices D(−5, −1), V(−1, −2), and W(−7, −4). ∆LPM

has vertices L(1, −5), P(2, −1), and M(4, −7). Graph both triangles on the coordinate plane. Then, using your drawing,

determine whether the triangles are congruent or not, providing a logical argument for your statement.

x

y

DV

W

Monday, February 6, 2012

Example 2∆DVW has vertices D(−5, −1), V(−1, −2), and W(−7, −4). ∆LPM

has vertices L(1, −5), P(2, −1), and M(4, −7). Graph both triangles on the coordinate plane. Then, using your drawing,

determine whether the triangles are congruent or not, providing a logical argument for your statement.

x

y

DV

W

Monday, February 6, 2012

Example 2∆DVW has vertices D(−5, −1), V(−1, −2), and W(−7, −4). ∆LPM

has vertices L(1, −5), P(2, −1), and M(4, −7). Graph both triangles on the coordinate plane. Then, using your drawing,

determine whether the triangles are congruent or not, providing a logical argument for your statement.

x

y

DV

WL

Monday, February 6, 2012

Example 2∆DVW has vertices D(−5, −1), V(−1, −2), and W(−7, −4). ∆LPM

has vertices L(1, −5), P(2, −1), and M(4, −7). Graph both triangles on the coordinate plane. Then, using your drawing,

determine whether the triangles are congruent or not, providing a logical argument for your statement.

x

y

DV

WL

P

Monday, February 6, 2012

Example 2∆DVW has vertices D(−5, −1), V(−1, −2), and W(−7, −4). ∆LPM

has vertices L(1, −5), P(2, −1), and M(4, −7). Graph both triangles on the coordinate plane. Then, using your drawing,

determine whether the triangles are congruent or not, providing a logical argument for your statement.

x

y

DV

WL

P

M

Monday, February 6, 2012

Example 2∆DVW has vertices D(−5, −1), V(−1, −2), and W(−7, −4). ∆LPM

has vertices L(1, −5), P(2, −1), and M(4, −7). Graph both triangles on the coordinate plane. Then, using your drawing,

determine whether the triangles are congruent or not, providing a logical argument for your statement.

x

y

DV

WL

P

M

Monday, February 6, 2012

Example 2∆DVW has vertices D(−5, −1), V(−1, −2), and W(−7, −4). ∆LPM

has vertices L(1, −5), P(2, −1), and M(4, −7). Graph both triangles on the coordinate plane. Then, using your drawing,

determine whether the triangles are congruent or not, providing a logical argument for your statement.

x

y

DV

WL

P

M

These are congruent. What possible ways could we show this?

Monday, February 6, 2012

Example 3Prove the following.

Prove: FEG ≅HIG

Given: EI ≅ FH, G is the midpoint of EI and FH

Monday, February 6, 2012

Example 3Prove the following.

Prove: FEG ≅HIG

Given: EI ≅ FH, G is the midpoint of EI and FH

1. EI ≅ FH , G is the midpoint of EI and FH

Monday, February 6, 2012

Example 3Prove the following.

Prove: FEG ≅HIG

Given: EI ≅ FH, G is the midpoint of EI and FH

1. EI ≅ FH , G is the midpoint of EI and FH 1. Given

Monday, February 6, 2012

Example 3Prove the following.

Prove: FEG ≅HIG

Given: EI ≅ FH, G is the midpoint of EI and FH

1. EI ≅ FH , G is the midpoint of EI and FH 1. Given

FG ≅ HG, EG ≅ IG2.

Monday, February 6, 2012

Example 3Prove the following.

Prove: FEG ≅HIG

Given: EI ≅ FH, G is the midpoint of EI and FH

1. EI ≅ FH , G is the midpoint of EI and FH 1. Given

FG ≅ HG, EG ≅ IG2. 2. Def. of midpoint

Monday, February 6, 2012

Example 3Prove the following.

Prove: FEG ≅HIG

Given: EI ≅ FH, G is the midpoint of EI and FH

1. EI ≅ FH , G is the midpoint of EI and FH 1. Given

FG ≅ HG, EG ≅ IG2. 2. Def. of midpoint

∠FGE ≅ ∠HGI3.

Monday, February 6, 2012

Example 3Prove the following.

Prove: FEG ≅HIG

Given: EI ≅ FH, G is the midpoint of EI and FH

1. EI ≅ FH , G is the midpoint of EI and FH 1. Given

FG ≅ HG, EG ≅ IG2. 2. Def. of midpoint

∠FGE ≅ ∠HGI3. 3. Vertical Angles

Monday, February 6, 2012

Example 3Prove the following.

Prove: FEG ≅HIG

Given: EI ≅ FH, G is the midpoint of EI and FH

1. EI ≅ FH , G is the midpoint of EI and FH 1. Given

FG ≅ HG, EG ≅ IG2. 2. Def. of midpoint

∠FGE ≅ ∠HGI3. 3. Vertical Angles

4. FEG ≅HIG

Monday, February 6, 2012

Example 3Prove the following.

Prove: FEG ≅HIG

Given: EI ≅ FH, G is the midpoint of EI and FH

1. EI ≅ FH , G is the midpoint of EI and FH 1. Given

FG ≅ HG, EG ≅ IG2. 2. Def. of midpoint

∠FGE ≅ ∠HGI3. 3. Vertical Angles

4. FEG ≅HIG 4. SAS

Monday, February 6, 2012

Check Your Understanding

Review p. 266 #1-4

Monday, February 6, 2012

Problem Set

Monday, February 6, 2012

Problem Set

p. 267 #5-19 odd, 31, 39, 41

"Doubt whom you will, but never yourself." - Christine Bovee

Monday, February 6, 2012