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Introduction to Geometry / LEVEL 1 · DIFFICULTY 1/5

Angles and Congruence

Use relationships to justify missing measures.

3 stages · 24 practice problems · two 6-question assessment forms

Choose an island to read its lesson.

  1. MINI QUEST GEO 1.1Straight AnglesRead the lesson
  2. MINI QUEST GEO 1.2Triangle Angle ReasoningRead the lesson
  3. MINI QUEST GEO 1.3Corresponding PartsRead the lesson
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STAGE GEO 1.1 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Straight Angles

Goal: Understand and apply straight angles.

Before you begin: Fractions, ratios, basic equations and square roots.

Understand the idea

Adjacent angles on a straight line partition a half-turn. Their measures add to 180°, even if the diagram is tilted or drawn imperfectly.

a+b=180°

Choose and carry out a method

Identify the common vertex and the straight outer rays. Subtract the known angle from 180; keep the degree unit in your reasoning.

Check the reasoning

Add both angles again. Each nondegenerate angle must be between 0° and 180°.

WORKED EXAMPLE 1

Two adjacent angles form a straight line. One is 24°. Find the other angle in degrees.

  1. A straight angle is 180 degrees.
  2. The missing angle is 180-24.
  3. It measures 156°, and the pair adds to 180°.

156

WORKED EXAMPLE 2

Two adjacent angles form a straight line. One is 27°. Find the other angle in degrees.

  1. A straight angle is 180 degrees.
  2. The missing angle is 180-27.
  3. It measures 153°, and the pair adds to 180°.

153

Common pitfalls

Possible mix-up: Every adjacent pair adds to 180°.

Only a pair whose outer rays form a straight line does.

Possible mix-up: A correct numerical answer alone explains the method.

State the governing relationship and check the conditions described above.

Explain it to yourself

How would the answer change if the outer rays made a right angle?

Preview the eight practice prompts
  1. Two adjacent angles form a straight line. One is 33°. Find the other angle in degrees.
  2. Two adjacent angles form a straight line. One is 36°. Find the other angle in degrees.
  3. Two adjacent angles form a straight line. One is 39°. Find the other angle in degrees.
  4. Two adjacent angles form a straight line. One is 42°. Find the other angle in degrees.
  5. Two adjacent angles form a straight line. One is 45°. Find the other angle in degrees.
  6. Two adjacent angles form a straight line. One is 48°. Find the other angle in degrees.
  7. A folding ruler opens to a straight 180° line. A brace divides that opening into 51° and one other angle. Find the other angle in degrees. New context
  8. A folding ruler opens to a straight 180° line. A brace divides that opening into 54° and one other angle. Find the other angle in degrees. New context
Open stage GEO 1.1 in the student workspace →

STAGE GEO 1.2 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Triangle Angle Reasoning

Useful preparation: Straight Angles

Goal: Understand and apply triangle angle reasoning.

Before you begin: Straight Angles

Understand the idea

Draw a line through one vertex parallel to the opposite side. Alternate interior angles fit the three triangle angles into a straight angle, explaining the 180° sum.

A+B+C=180°

Choose and carry out a method

Name all three interior angles. Subtract the two known measures from 180, and avoid using an exterior angle as an interior one.

Check the reasoning

All three measures must be positive. A sketch suggests relationships but does not prove the measurements.

WORKED EXAMPLE 1

Triangle ABC has angles A=21° and B=32°. Find angle C in degrees.

  1. The interior angles of a Euclidean triangle sum to 180 degrees.
  2. C=180-21-32.
  3. C=127°. All three angles are positive and their sum is 180°.

127

WORKED EXAMPLE 2

Triangle ABC has angles A=22° and B=34°. Find angle C in degrees.

  1. The interior angles of a Euclidean triangle sum to 180 degrees.
  2. C=180-22-34.
  3. C=124°. All three angles are positive and their sum is 180°.

124

Common pitfalls

Possible mix-up: An exterior angle belongs in the interior sum.

Use its supplementary interior angle instead.

Possible mix-up: A correct numerical answer alone explains the method.

State the governing relationship and check the conditions described above.

Explain it to yourself

Explain the angle-sum rule using a parallel line.

Preview the eight practice prompts
  1. Triangle ABC has angles A=24° and B=38°. Find angle C in degrees.
  2. Triangle ABC has angles A=25° and B=40°. Find angle C in degrees.
  3. Triangle ABC has angles A=26° and B=42°. Find angle C in degrees.
  4. Triangle ABC has angles A=27° and B=44°. Find angle C in degrees.
  5. Triangle ABC has angles A=28° and B=46°. Find angle C in degrees.
  6. Triangle ABC has angles A=29° and B=48°. Find angle C in degrees.
  7. A triangular roof frame has two interior angles of 30° and 50°. What is the third interior angle in degrees? New context
  8. A triangular roof frame has two interior angles of 31° and 52°. What is the third interior angle in degrees? New context
Open stage GEO 1.2 in the student workspace →

STAGE GEO 1.3 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Corresponding Parts

Useful preparation: Triangle Angle Reasoning

Goal: Understand and apply corresponding parts.

Before you begin: Triangle Angle Reasoning

Understand the idea

Congruent triangles have equal corresponding sides and angles. The written vertex order records the correspondence, so ABC≅DEF means A matches D, B matches E and C matches F.

ABC≅DEF ⇒ ∠C=∠F

Choose and carry out a method

Write the vertex matching first. Transfer known angles to matching vertices, then apply the triangle angle sum for a missing angle.

Check the reasoning

Rotation or reflection changes appearance but preserves congruence. Match labels, not visual position.

WORKED EXAMPLE 1

Congruent triangles ABC and DEF have corresponding vertices A↔D, B↔E, C↔F. If A=29° and B=42°, find F in degrees.

  1. Congruence preserves corresponding angles as well as lengths.
  2. The remaining angle in the first triangle is 180-29-42.
  3. Its matching angle is also 109°; vertex correspondence identifies the correct angle.

109

WORKED EXAMPLE 2

Congruent triangles ABC and DEF have corresponding vertices A↔D, B↔E, C↔F. If A=30° and B=43°, find F in degrees.

  1. Congruence preserves corresponding angles as well as lengths.
  2. The remaining angle in the first triangle is 180-30-43.
  3. Its matching angle is also 107°; vertex correspondence identifies the correct angle.

107

Common pitfalls

Possible mix-up: The leftmost vertices must correspond.

Correspondence comes from the stated order, regardless of orientation.

Possible mix-up: A correct numerical answer alone explains the method.

State the governing relationship and check the conditions described above.

Explain it to yourself

What additional information would prove congruence instead of merely assuming it?

Preview the eight practice prompts
  1. Congruent triangles ABC and DEF have corresponding vertices A↔D, B↔E, C↔F. If A=32° and B=45°, find F in degrees.
  2. Congruent triangles ABC and DEF have corresponding vertices A↔D, B↔E, C↔F. If A=33° and B=46°, find F in degrees.
  3. Congruent triangles ABC and DEF have corresponding vertices A↔D, B↔E, C↔F. If A=34° and B=47°, find F in degrees.
  4. Congruent triangles ABC and DEF have corresponding vertices A↔D, B↔E, C↔F. If A=35° and B=48°, find F in degrees.
  5. Congruent triangles ABC and DEF have corresponding vertices A↔D, B↔E, C↔F. If A=36° and B=49°, find F in degrees.
  6. Congruent triangles ABC and DEF have corresponding vertices A↔D, B↔E, C↔F. If A=37° and B=50°, find F in degrees.
  7. Two triangular panels are identical rigid copies. The first has angles 38° and 51°. A hinge on the second is at the vertex corresponding to the remaining angle. What angle does it have, in degrees? New context
  8. Two triangular panels are identical rigid copies. The first has angles 39° and 52°. A hinge on the second is at the vertex corresponding to the remaining angle. What angle does it have, in degrees? New context
Open stage GEO 1.3 in the student workspace →
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