Make learning comfortable

Visual experienceAnimationRead-aloud speedChanging style, motion, or reading speed keeps your current question, answers, homework, and saved progress. The school band still chooses your task layout. Saved to your learner profile across devices. All lessons and help stay available.

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.

MINI QUEST GEO 1.1Straight AnglesRead the lesson
MINI QUEST GEO 1.2Triangle Angle ReasoningRead the lesson
MINI QUEST GEO 1.3Corresponding PartsRead the lesson
LEVEL CHECKCastle challenge
6 questions across this level.
Sign in for the level check →

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.

NOTICE → MARK → REASON

Two angles on a straight line

Mark the known angle and the missing angle in different colors. This is a separate learning diagram; it does not supply the values for your practice question.

Highlight parts and add your information

Choose a part here, or select it in the diagram. Mark what you know, what you need, or a reason. These are your scratch notes; the website does not grade or save them.

One small move: Select a part above. Find its given measurement in the question, then label that part below.

Part of the diagramWhat does your mark mean?Length, angle or reasoning

Use the problem’s stated values. A drawing can look shorter after rotation even when the actual length stays the same. Changing the model or reloading clears these notes.

Understand the idea

Adjacent angles on a straight line partition a half-turn. Their measures add to $\displaystyle 180^{\circ}$ , even if the diagram is tilted or drawn imperfectly.

\displaystyle a + b = 180^{\circ}

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 $\displaystyle 0^{\circ}$ and $\displaystyle 180^{\circ}$ .

WORKED EXAMPLE 1

Two adjacent angles form a straight line. One is $\displaystyle 24^{\circ}$ . Find the other angle in degrees.

A straight angle is 180 degrees.
The missing angle is $\displaystyle 180 - 24$ .
It measures $\displaystyle 156^{\circ}$ , and the pair adds to $\displaystyle 180^{\circ}$ .

156

WORKED EXAMPLE 2

Two adjacent angles form a straight line. One is $\displaystyle 27^{\circ}$ . Find the other angle in degrees.

A straight angle is 180 degrees.
The missing angle is $\displaystyle 180 - 27$ .
It measures $\displaystyle 153^{\circ}$ , and the pair adds to $\displaystyle 180^{\circ}$ .

153

Common pitfalls

Possible mix-up: Every adjacent pair adds to $\displaystyle 180^{\circ}$ .

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

Problem 1
Two adjacent angles form a straight line. One is $\displaystyle 33^{\circ}$ . Find the other angle in degrees.
Practise this exact problem →
Problem 2
Two adjacent angles form a straight line. One is $\displaystyle 36^{\circ}$ . Find the other angle in degrees.
Practise this exact problem →
Problem 3
Two adjacent angles form a straight line. One is $\displaystyle 39^{\circ}$ . Find the other angle in degrees.
Practise this exact problem →
Problem 4
Two adjacent angles form a straight line. One is $\displaystyle 42^{\circ}$ . Find the other angle in degrees.
Practise this exact problem →
Problem 5
Two adjacent angles form a straight line. One is $\displaystyle 45^{\circ}$ . Find the other angle in degrees.
Practise this exact problem →
Problem 6
Two adjacent angles form a straight line. One is $\displaystyle 48^{\circ}$ . Find the other angle in degrees.
Practise this exact problem →
Problem 7New context
A folding ruler opens to a straight $\displaystyle 180^{\circ}$ line. A brace divides that opening into $\displaystyle 51^{\circ}$ and one other angle. Find the other angle in degrees.
Practise this exact problem →
Problem 8New context
A folding ruler opens to a straight $\displaystyle 180^{\circ}$ line. A brace divides that opening into $\displaystyle 54^{\circ}$ and one other angle. Find the other angle in degrees.
Practise this exact problem →

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 $\displaystyle 180^{\circ}$ sum.

\displaystyle A + B + C = 180^{\circ}

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 $\displaystyle A = 21^{\circ}$ and $\displaystyle B = 32^{\circ}$ . Find angle C in degrees.

The interior angles of a Euclidean triangle sum to 180 degrees.
$\displaystyle C = 180 - 21 - 32$ .
$\displaystyle C = 127^{\circ}$ . All three angles are positive and their sum is $\displaystyle 180^{\circ}$ .

127

WORKED EXAMPLE 2

Triangle ABC has angles $\displaystyle A = 22^{\circ}$ and $\displaystyle B = 34^{\circ}$ . Find angle C in degrees.

The interior angles of a Euclidean triangle sum to 180 degrees.
$\displaystyle C = 180 - 22 - 34$ .
$\displaystyle C = 124^{\circ}$ . All three angles are positive and their sum is $\displaystyle 180^{\circ}$ .

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

Problem 1
Triangle ABC has angles $\displaystyle A = 24^{\circ}$ and $\displaystyle B = 38^{\circ}$ . Find angle C in degrees.
Practise this exact problem →
Problem 2
Triangle ABC has angles $\displaystyle A = 25^{\circ}$ and $\displaystyle B = 40^{\circ}$ . Find angle C in degrees.
Practise this exact problem →
Problem 3
Triangle ABC has angles $\displaystyle A = 26^{\circ}$ and $\displaystyle B = 42^{\circ}$ . Find angle C in degrees.
Practise this exact problem →
Problem 4
Triangle ABC has angles $\displaystyle A = 27^{\circ}$ and $\displaystyle B = 44^{\circ}$ . Find angle C in degrees.
Practise this exact problem →
Problem 5
Triangle ABC has angles $\displaystyle A = 28^{\circ}$ and $\displaystyle B = 46^{\circ}$ . Find angle C in degrees.
Practise this exact problem →
Problem 6
Triangle ABC has angles $\displaystyle A = 29^{\circ}$ and $\displaystyle B = 48^{\circ}$ . Find angle C in degrees.
Practise this exact problem →
Problem 7New context
A triangular roof frame has two interior angles of $\displaystyle 30^{\circ}$ and $\displaystyle 50^{\circ}$ . What is the third interior angle in degrees?
Practise this exact problem →
Problem 8New context
A triangular roof frame has two interior angles of $\displaystyle 31^{\circ}$ and $\displaystyle 52^{\circ}$ . What is the third interior angle in degrees?
Practise this exact problem →

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 $\displaystyle A = 29^{\circ}$ and $\displaystyle B = 42^{\circ}$ , find F in degrees.

Congruence preserves corresponding angles as well as lengths.
The remaining angle in the first triangle is $\displaystyle 180 - 29 - 42$ .
Its matching angle is also $\displaystyle 109^{\circ}$ ; 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 $\displaystyle A = 30^{\circ}$ and $\displaystyle B = 43^{\circ}$ , find F in degrees.

Congruence preserves corresponding angles as well as lengths.
The remaining angle in the first triangle is $\displaystyle 180 - 30 - 43$ .
Its matching angle is also $\displaystyle 107^{\circ}$ ; 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

Problem 1
Congruent triangles ABC and DEF have corresponding vertices A↔D, B↔E, C↔F. If $\displaystyle A = 32^{\circ}$ and $\displaystyle B = 45^{\circ}$ , find F in degrees.
Practise this exact problem →
Problem 2
Congruent triangles ABC and DEF have corresponding vertices A↔D, B↔E, C↔F. If $\displaystyle A = 33^{\circ}$ and $\displaystyle B = 46^{\circ}$ , find F in degrees.
Practise this exact problem →
Problem 3
Congruent triangles ABC and DEF have corresponding vertices A↔D, B↔E, C↔F. If $\displaystyle A = 34^{\circ}$ and $\displaystyle B = 47^{\circ}$ , find F in degrees.
Practise this exact problem →
Problem 4
Congruent triangles ABC and DEF have corresponding vertices A↔D, B↔E, C↔F. If $\displaystyle A = 35^{\circ}$ and $\displaystyle B = 48^{\circ}$ , find F in degrees.
Practise this exact problem →
Problem 5
Congruent triangles ABC and DEF have corresponding vertices A↔D, B↔E, C↔F. If $\displaystyle A = 36^{\circ}$ and $\displaystyle B = 49^{\circ}$ , find F in degrees.
Practise this exact problem →
Problem 6
Congruent triangles ABC and DEF have corresponding vertices A↔D, B↔E, C↔F. If $\displaystyle A = 37^{\circ}$ and $\displaystyle B = 50^{\circ}$ , find F in degrees.
Practise this exact problem →
Problem 7New context
Two triangular panels are identical rigid copies. The first has angles $\displaystyle 38^{\circ}$ and $\displaystyle 51^{\circ}$ . A hinge on the second is at the vertex corresponding to the remaining angle. What angle does it have, in degrees?
Practise this exact problem →
Problem 8New context
Two triangular panels are identical rigid copies. The first has angles $\displaystyle 39^{\circ}$ and $\displaystyle 52^{\circ}$ . A hinge on the second is at the vertex corresponding to the remaining angle. What angle does it have, in degrees?
Practise this exact problem →

Open stage GEO 1.3 in the student workspace →

Browse another level →

Angles and Congruence

Straight Angles

Two angles on a straight line

Understand the idea

Choose and carry out a method

Check the reasoning

Two adjacent angles form a straight line. One is 24∘\displaystyle 24^{\circ}24∘. Find the other angle in degrees.

Two adjacent angles form a straight line. One is 27∘\displaystyle 27^{\circ}27∘. Find the other angle in degrees.

Common pitfalls

Explain it to yourself

Triangle Angle Reasoning

Understand the idea

Choose and carry out a method

Check the reasoning

Triangle ABC has angles A=21∘\displaystyle A = 21^{\circ}A=21∘ and B=32∘\displaystyle B = 32^{\circ}B=32∘. Find angle C in degrees.

Triangle ABC has angles A=22∘\displaystyle A = 22^{\circ}A=22∘ and B=34∘\displaystyle B = 34^{\circ}B=34∘. Find angle C in degrees.

Common pitfalls

Explain it to yourself

Corresponding Parts

Understand the idea

Choose and carry out a method

Check the reasoning

Congruent triangles ABC and DEF have corresponding vertices A↔D, B↔E, C↔F. If A=29∘\displaystyle A = 29^{\circ}A=29∘ and B=42∘\displaystyle B = 42^{\circ}B=42∘, find F in degrees.

Congruent triangles ABC and DEF have corresponding vertices A↔D, B↔E, C↔F. If A=30∘\displaystyle A = 30^{\circ}A=30∘ and B=43∘\displaystyle B = 43^{\circ}B=43∘, find F in degrees.

Common pitfalls

Explain it to yourself

Two adjacent angles form a straight line. One is $\displaystyle 24^{\circ}$ . Find the other angle in degrees.

Two adjacent angles form a straight line. One is $\displaystyle 27^{\circ}$ . Find the other angle in degrees.

Triangle ABC has angles $\displaystyle A = 21^{\circ}$ and $\displaystyle B = 32^{\circ}$ . Find angle C in degrees.

Triangle ABC has angles $\displaystyle A = 22^{\circ}$ and $\displaystyle B = 34^{\circ}$ . Find angle C in degrees.

Congruent triangles ABC and DEF have corresponding vertices A↔D, B↔E, C↔F. If $\displaystyle A = 29^{\circ}$ and $\displaystyle B = 42^{\circ}$ , find F in degrees.

Congruent triangles ABC and DEF have corresponding vertices A↔D, B↔E, C↔F. If $\displaystyle A = 30^{\circ}$ and $\displaystyle B = 43^{\circ}$ , find F in degrees.