Trigonometric geometry and proofs

Which angle number represents an angle vertical to ∠FHC?

Angle number 6 is vertical to ∠FHC.

Which angle number represents the linear pair of ∠UYV?

Angle number 5 is the linear pair of ∠UYV.

Which angle number represents ∠DGB?

 ∠DGB is angle number 2.

Which angle number represents the linear pair of ∠PSL?

Angle number 5 is the linear pair of ∠PSL.

Find the measure of the missing angle.

36

Find the measure of the missing angle.

a = 87°

Find the measure of the missing angles.

b = 50°

c = 130°

Given m∥n, find the value of x.

x = ___

x = 26∘

Find the measure of the missing angle.

a = 71°

Given m∥n, find the value of x.

x = 80∘

Given m∥n, find the value of x.

x = 52∘

Given m∥n, find the value of x.

x = 30∘

Fill out the rest of the statement/reason table to prove the triangles are similar

Step	Statement	Reason
1	BC≅AD BC∥AD	Given
2	AC≅CA	Reflexive Property
3	∠ACB≅∠DAC	Alternate interior angles are congruent when lines are parallel.
4	△ABC≅△CDA	SAS

Fill out the rest of the statement/reason table to prove the triangles are similar

Step	Statement	Reason
1	BC≅AD AB≅CD	Given
2	AC≅CA	Reflexive Property
3	△ABC≅△CDA	SSS

Fill out the rest of the statement/reason table to prove the triangles are similar

Step	Statement	Reason
1	DE≅DF AD≅DC	Given
2	∠ADE≅∠CDF	Vertical angles are congruent
3	△ADE≅△CDF	SAS

Fill out the rest of the statement/reason table to prove the triangles are similar

Step	Statement	Reason
1	AC and DB bisect each other	Given
2	AE≅EC	A segment bisector divides a segment into two congruent segments
3	BE≅ED	A segment bisector divides a segment into two congruent segments
4	∠AEB≅∠CED	Vertical angles are congruent
5	△ABE≅△CDE	SAS

Fill out the rest of the statement/reason table to prove the triangles are similar

Step	Statement	Reason
1	∠B≅∠D BC∥AD	Given
2	∠BCA≅∠DAC	Alternate interior angles are congruent when lines are parallel.
3	CA≅AC	Reflexive Property
4	△ABC≅△CDA	AAS

Fill out the rest of the statement/reason table to prove the triangles are similar

Step	Statement	Reason
1	AC bisects BD ∠CBE≅∠EDA	Given
2	BE≅ED	A segment bisector divides a segment into two congruent segments
3	∠AED≅∠CEB	Vertical angles are congruent
4	△BEC≅△DEA	ASA

Fill out the rest of the statement/reason table to prove the triangles are similar

Step	Statement	Reason
1	∠B≅∠D AD∥BC	Given
2	∠BCA≅∠DAC	Alternate interior angles are congruent when lines are parallel.
3	CA≅AC	Reflexive Property
4	△ABC≅△CDA	AAS

Fill out the rest of the statement/reason table to prove the triangles are similar

Step	Statement	Reason
1	DB bisects ∠ABC ∠A≅∠C	Given
2	∠ABD≅∠CBD	An angle bisector divides an angle into two congruent angles
3	BD≅DB	Reflexive Property
4	△ABD≅△CBD	AAS

Determine the relationship between the two triangles and whether or not they can be proven to be congruent.

The two triangles are related by _____ , so the triangles ___.

The two triangles are related by Side-Side-Angle (SSA), so the triangles CANNOT be proven congruent.

Determine the relationship between the two triangles and whether or not they can be proven to be congruent.

The two triangles are related by _____ , so the triangles ___.

The two triangles are related by Angle-Angle-Angle (AAA), so the triangles CANNOT be proven congruent.

Determine the relationship between the two triangles and whether or not they can be proven to be congruent.

The two triangles are related by _____ , so the triangles ___.

The two triangles are related by Angle-Angle-Side (AAS), so the triangles can be proven congruent.

Determine the relationship between the two triangles and whether or not they can be proven to be congruent.

The two triangles are related by _____ , so the triangles ___.

The two triangles are related by Side-Angle-Side (SAS), so the triangles can be proven congruent.

Determine the relationship between the two triangles and whether or not they can be proven to be congruent.

The two triangles are related by _____ , so the triangles ___.

The two triangles are related by Angle-Angle-Side (AAS), so the triangles can be proven congruent.

Based on the given diagram, complete the flowchart proof below.

Reason: Given,

Reason: Given,

Reason: Vertical angles are congruent,

Reason: ASA

Given BD bisects AC, complete the flowchart proof below.

Reason: A segment bisector divides a segment into two congruent segments,

Reason: Given,

Reason: Vertical angles are congruent,

Reason: AAS

Based on the given diagram, complete the flowchart proof below.

Reason: Given,

Reason: Given,

Reason: Reflexive Property,

Reason: SAS,

Given AC is the angle bisector of ∠BAD and ∠BCD, complete the flowchart proof below.

Reason: An angle bisector divides an angle into two congruent angles,

Reason: An angle bisector divides an angle into two congruent angles,

Reason: Given,

Reason: AAS

Given AB∥DC, complete the flowchart proof below.

Reason: Alternate interior angles are congruent when lines are parallel.

Reason: Vertical angles are congruent.

Reason: Given

Reason: AAS

Given AC bisects ∠BCD, complete the flowchart proof below. Note that the last statement and reason have both been filled in for you.

1. AC bisects ∠BCD Reason: Given

2. ∠BCA ≅ ∠DCA Reason: An angle bisector divides an angle into two congruent angles

3. BC ≅ DC Reason: Given

4. AC ≅ AC Reason: Reflexive Property

5. △ABC≅△ADC Reason: SAS

Given AC is the angle bisector of ∠BAD, complete the flowchart proof below. Note that the last statement and reason have both been filled in for you.

1. AC bisects ∠BAD Reason: Given

2. ∠BAC ≅ ∠DAC Reason: An angle bisector divides an angle into two congruent angles

3. AB ≅ AD Reason: Given

4. AC ≅ AC Reason: Reflexive Property

5. △ABC≅△ADC Reason: SAS

Given AB∥DC and BC∥AD, complete the flowchart proof below. Note that the last statement and reason have both been filled in for you

1. AB ∥ DC Reason: Given

2. ∠BAC ≅ ∠DCA Reason: Parallel lines cut by a transversal form congruent alternate interior angles

3. BC ∥ AD Reason: Given

4. ∠BCA ≅ ∠DAC Reason: Parallel lines cut by a transversal form congruent alternate interior angles

5. AB ≅ CD Reason: Given

6. △ABC≅△CDA Reason: AAS

Given AB≅CB, prove △ABE≅△CBD by filling out the flowchart below.

1. ∠A ≅ ∠C Reason: `Given

2. AB ≅ CB Reason: Given

3. ∠B ≅ ∠B Reason: Reflexive Property

4. △ABE≅△CBD Reason: ASA

Given BD bisects AC and AC⊥BD, prove △ABD≅△CBD by filling out the flowchart below.

1. AC ⊥ BD Reason: Given

2. ∠ADB and ∠CDB are right angles Reason: Perpendicular lines form right angles

3. ∠ADB ≅ ∠CDB Reason: All right angles are congruent

4. BD bisects AC Reason: Given

5. AD ≅ CD Reason: A segment bisector divides a segment into two congruent segment

6. BD ≅ BD Reason: Reflexive Property

7. △ABD≅△CBD Reason: SAS

Given BD is the angle bisector of ∠ABC, prove △ABD≅△CBD by filling out the flowchart below.

1. BD bisects ∠ABC Reason: Given

2. ∠ABD ≅ ∠CBD Reason: An angle bisector divides an angle into two congruent angles

3. ∠ADB ≅ ∠CDB Reason: Given

4. BD ≅ BD Reason: Reflexive Property

5. △ABD≅△CBD Reason: ASA

Given AC⊥BD, prove △ABD≅△CBD by filling out the flowchart below.

1. AC ⊥ BD Reason: Given

2. ∠ADB and ∠CDB are right angles Reason: Perpendicular lines form right angles

3. ∠ADB ≅ ∠CDB Reason: All right angles are congruent

4. ∠ABD ≅ ∠CBD Reason: Given

5. BD ≅ BD Reason: Reflexive Property

6. △ABD≅△CBD Reason: ASA

Fill out the rest of the statement/reason table to prove the triangles are similar

Given: ∠A≅∠C and AE≅CF.

Prove: AD≅CD.

Step	Statement	Reason
1	AE≅EC BE≅ED	Given
2	∠AEB≅∠CED	Vertical angles are congruent
3	△AEB≅△CED	SAS
4	AB≅DC	Corresponding Parts of Congruent Triangles are Congruent (CPCTC)

Given: ∠A≅∠C and AE≅CF.

Prove: AD≅CD.

Fill out the rest of the statement/reason table to prove the triangles are similar

Step	Statement	Reason
1	∠A≅∠C AE≅CF	Given
2	∠EDA≅∠FDC	Vertical angles are congruent
3	△DFC≅△DEA	AAS
4	AD≅CD	Corresponding Parts of Congruent Triangles are Congruent (CPCTC)

Given: BD and AC bisect each other.

Prove: AB∥CD.

Fill out the rest of the statement/reason table to prove the triangles are similar

Step	Statement	Reason
1	BD and AC bisect each other	Given
2	AE≅EC	A segment bisector divides a segment into two congruent segments
3	DE≅EB	A segment bisector divides a segment into two congruent segments
4	∠AEB≅∠CED	Vertical angles are congruent
5	△AEB≅△CED	SAS
6	AB≅CD	Corresponding Parts of Congruent Triangles are Congruent (CPCTC)
7	∠EAB≅∠ECD	Corresponding Parts of Congruent Triangles are Congruent (CPCTC)
8	∠EBA≅∠EDC	Corresponding Parts of Congruent Triangles are Congruent (CPCTC)
9	AB∥CD	If two lines form congruent alternate interior angles, then the two lines are parallel

Given: AD≅AE and △CFE≅△BFD.

Prove: △BAE≅△CAD.

Fill out the rest of the statement/reason table to prove the triangles are similar

Step	Statement	Reason
1	AD≅AE △CFE≅△BFD	Given
2	∠BAE≅∠CAD	Reflexive Property
3	∠DBF≅∠ECF	Corresponding Parts of Congruent Triangles are Congruent (CPCTC)
4	△ACD≅△ABE	AAS

A side of the triangle below has been extended to form an exterior angle of 157°. Find the value of x.

x = 23

A side of the triangle below has been extended to form an exterior angle of 129°. Find the value of x.

x = 51

A side of the triangle below has been extended to form an exterior angle of 121°. Find the value of x.

x = 31

In ΔSTU,  SU is extended through point U to point V,  m∠UST=(x+5)∘,  m∠STU=(2x+16)∘,   and m∠TUV=(6x−9)∘.   Find m∠STU.

m∠STU=36∘

In ΔFGH,  FH is extended through point H to point I,  m∠HFG=(3x+15)∘,  m∠GHI=(6x−6)∘,   and m∠FGH=(x+1)∘.   Find m∠HFG.

m∠HFG=48∘

In ΔPQR,  PR is extended through point R to point S,  m∠PQR=(3x+8)∘,  m∠QRS=(9x−17)∘,   and m∠RPQ=(3x+2)∘.   Find m∠QRS.

m∠QRS=64∘

Determine the type of triangle that is drawn below.

1. Isosceles Acute

2. Scalene Obtuse

3. Equilateral

4. Scalene Right

5. Isosceles Right

6. Scalene Acute

7. Isosceles Obtuse

5. Isosceles Right

Determine the type of triangle that is drawn below.

1. Isosceles Acute

2. Scalene Obtuse

3. Equilateral

4. Scalene Right

5. Isosceles Right

6. Scalene Acute

7. Isosceles Obtuse

5. Isosceles Right

Determine the type of triangle that is drawn below.

1. Isosceles Acute

2. Scalene Obtuse

3. Equilateral

4. Scalene Right

5. Isosceles Right

6. Scalene Acute

7. Isosceles Obtuse

6. Scalene Acute

1. Isosceles Acute

2. Scalene Obtuse

3. Equilateral

4. Scalene Right

5. Isosceles Right

6. Scalene Acute

7. Isosceles Obtuse

2. Scalene Obtuse

Which of the following sets of numbers could represent the three sides of a triangle?

a. {10,21,28}

b. {9,19,28}

c. {9,20,30}

d. {10,25,35}

a. {10,21,28}

Which of the following sets of numbers could represent the three sides of a triangle?

a. {15,28,45}

b. {5,15,20}

c. {10,16,24}

d. {11,24,37}

c. {10,16,24}

Which of the following sets of numbers could represent the three sides of a triangle?

a. {5,20,25}

b. {6,21,27}

c. {6,12,15}

d. {7,21,29}

c. {6,12,15}

Which of the following sets of numbers could represent the three sides of a triangle?

a. {12,15,28}

b. {10,25,36}

c. {10,21,29}

d. {15,27,42}

c. {10,21,29}

Which of the following sets of numbers could represent the three sides of a triangle?

a. {11,22,32}

b. {8,22,31}

c. {13,17,30}

d. {10,23,33}

a. {11,22,32}

In ΔXYZ, YZ = 7, ZX = 15, and XY = 13. Which statement about the angles of ΔXYZ must be true?

a. m∠X<m∠Y<m∠Z

b. m∠X<m∠Z<m∠Y

c. m∠Z<m∠Y<m∠X

d. m∠Y<m∠X<m∠Z

e. m∠Y<m∠Z<m∠X

f. m∠Z<m∠X<m∠Y

b. m∠X<m∠Z<m∠Y

In ΔQRS, m∠Q = 86° and m∠R = 15°. Which statement about the sides of ΔQRS must be true?

a. QR>RS>SQ

b. RS>QR>SQ

c. SQ>RS>QR

d. SQ>QR>RS

e. RS>SQ>QR

f. QR>SQ>RS

b. RS>QR>SQ