3.5 Understanding angles, congruence, and similarity

Achievable Praxis Core: Math (5733)

3. Algebra and geometry

Our Praxis Core: Math course is currently in development and is a work-in-progress.

Understanding angles, congruence, and similarity

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This section covers common angle relationships, how to use angle rules to find unknown measures, and how to determine whether triangles are congruent or similar. You’ll also see how scaling similar figures affects perimeter, area, and volume - and why those changes involve $k$ , $k^{2}$ , and $k^{3}$ .

Definitions

Vertical angles: Angles opposite each other when two lines intersect; always congruent.
Complementary angles: Two angles whose measures add to $9 0^{\circ}$ .
Supplementary angles: Two angles whose measures add to $18 0^{\circ}$ .
Similar figures: Figures that have the same shape but different sizes, with all corresponding angles equal and side lengths proportional.
Scale factor $k$: The ratio of corresponding side lengths between two similar figures.

Angle relationships

When two lines intersect - or when a transversal crosses parallel lines - certain angle pairs always have predictable relationships. Once you recognize the pattern, you can set up an equation and solve for unknown angle measures.

Angles formed by parallel lines and a transversal:
- Corresponding angles are in the same position on each parallel line and are always equal.
- Alternate interior angles are inside the parallel lines on opposite sides of the transversal and are always equal.
- Alternate exterior angles are outside the parallel lines on opposite sides of the transversal and are always equal.
- Same-side interior angles are inside the parallel lines on the same side of the transversal and always add up to $18 0^{\circ}$ (supplementary).

Example: Supplementary angles

Two same-side interior angles formed by a transversal cutting two parallel lines measure $3 x$ and $2 x + 20$ . Because same-side interior angles are supplementary, their measures add to $18 0^{\circ}$ .

Transversal angle measurements

Set up the equation:
$3 x + (2 x + 20) = 180$

Combine like terms:
$5 x + 20 = 180$

Subtract $20$ :
$5 x = 160$

Divide by $5$ :
$x = 32$

The angles are $3 (32) = 9 6^{\circ}$ and $2 (32) + 20 = 8 4^{\circ}$ .

Answer: $x = 32$ ; the two angles are $9 6^{\circ}$ and $8 4^{\circ}$

Congruent triangles

Two triangles are congruent when they match exactly in both shape and size - all corresponding sides and angles are equal. Recall that the three interior angles of any triangle sum to $18 0^{\circ}$ , which lets you find a missing angle whenever two are known.

You could check all six parts, but it’s usually faster to use one of the standard congruence tests:

SSS: All three sides are equal.
SAS: Two sides and the angle between them are equal.
ASA: Two angles and the included side (the side between those two angles) are equal.
AAS: Two angles and a non-included side (a side not between the two given angles) are equal.
HL: The hypotenuse and one leg in a right triangle are equal.

Common pitfalls

ASA vs AAS: The difference is which side you know. In ASA the known side is between the two given angles; in AAS it is not between them. Mixing these up is one of the most frequent errors on this topic.
Corresponding letters matter in similar figures. When figures are named (e.g., pentagon $A BC D E \sim$ pentagon $PQRST$ ), the letter order tells you which sides correspond. Writing $\frac{A B}{QR}$ instead of $\frac{A B}{PQ}$ is a common proportion error.

Below are examples of two congruence configurations.

Example: Determining congruence (AAS)

Given triangle $A BC$ and triangle $D EF$ with the following:

$∠ A = 2 5^{\circ}$ , $∠ C = 8 3^{\circ}$ , and $A B = 3$ in triangle $A BC$

$∠ D = 2 5^{\circ}$ , $∠ F = 8 3^{\circ}$ , and $D E = 3$ in triangle $D EF$

Are the triangles congruent? If so, state why.

Triangle ABC

Triangle DEF

We have two pairs of equal angles: $∠ A = ∠ D = 2 5^{\circ}$ and $∠ C = ∠ F = 8 3^{\circ}$ . The known side $A B = D E = 3$ connects vertices $A$ and $B$ . Because $A B$ does not touch $∠ C$ (or $∠ F$ ), it is not between the two given angles - this is the non-included side condition for AAS.

Therefore $△ A BC ≅ △ D EF$ by AAS, which means all corresponding sides are equal: $BC = EF$ and $C A = F D$ .

Answer: Yes, congruent by AAS

Similar figures and scaling

Two figures are similar if they have the same shape but different sizes. In similar figures:

All corresponding angles are equal.
All corresponding side lengths are proportional by a constant scale factor, denoted $k$ .

When figures are similar, here’s how measurements change:

Angles remain unchanged.
Side lengths scale by a factor of $k$ .
Perimeter scales by $k$ , since each linear measurement is multiplied by $k$ .
Area scales by $k^{2}$ , because area depends on two dimensions.
Volume scales by $k^{3}$ , because volume depends on three dimensions.

When no diagram is provided, you can identify corresponding sides and angles by using the order of letters in the figure names. For example, if triangle $A BC$ is similar to triangle $D EF$ , then $∠ A ≅ ∠ D$ , $∠ B ≅ ∠ E$ , and $∠ C ≅ ∠ F$ ; side $A B$ corresponds to $D E$ , $BC$ to $EF$ , and $A C$ to $D F$ .

Example: Solving for a side in similar polygons

Pentagon $A BC D E$ is similar to pentagon $PQRST$ . If $A B = 6.4 cm$ , $PQ = 10 cm$ , and $BC = 8.1 cm$ , find the length of $QR$ .

Since the pentagons are similar, their corresponding sides are in proportion. $A B$ corresponds to $PQ$ and $BC$ corresponds to $QR$ , so:

$\frac{A B}{PQ} = \frac{BC}{QR} \Rightarrow \frac{6.4}{10} = \frac{8.1}{QR}$

Solve the proportion:

Cross-multiply: $6.4 \cdot QR = 10 \cdot 8.1$

Multiply: $6.4 QR = 81$

Divide: $QR = \frac{81}{6.4} \approx 12.66$

Answer: $12.66 cm$

Example: Scale factor with area and volume

A square with side $4$ has area $16$ . Scaling each side by $k = 3$ gives side $12$ and area $144$ . The ratio $\frac{144}{16} = 9 = 3^{2}$ confirms that area scales by $k^{2}$ . Likewise, scaling each dimension of a solid by $k = 3$ multiplies volume by $3^{3} = 27$ .

Square

Square scaled by 3

Answer: Area scales by $k^{2}$ ( $9 \times$ ); volume scales by $k^{3}$ ( $27 \times$ )

Vertical angles are congruent; linear pairs are supplementary.
Complementary angles sum to $9 0^{\circ}$ ; supplementary angles sum to $18 0^{\circ}$ .
Parallel lines cut by a transversal form congruent corresponding, alternate interior, and alternate exterior angles; same-side interior angles are supplementary.
Prove triangle congruence using SSS, SAS, ASA, AAS, or HL (SSA is invalid).
Similar figures have equal angles and proportional sides.
For scale factor $k$ , perimeter scales by $k$ , area by $k^{2}$ , and volume by $k^{3}$ .
Match corresponding parts using the letter order in figure names to solve similarity problems without diagrams.

Understanding angles, congruence, and similarity

Definitions

Vertical angles: Angles opposite each other when two lines intersect; always congruent.
Complementary angles: Two angles whose measures add to $9 0^{\circ}$ .
Supplementary angles: Two angles whose measures add to $18 0^{\circ}$ .
Similar figures: Figures that have the same shape but different sizes, with all corresponding angles equal and side lengths proportional.
Scale factor $k$: The ratio of corresponding side lengths between two similar figures.

Angle relationships

Angles formed by parallel lines and a transversal:
- Corresponding angles are in the same position on each parallel line and are always equal.
- Alternate interior angles are inside the parallel lines on opposite sides of the transversal and are always equal.
- Alternate exterior angles are outside the parallel lines on opposite sides of the transversal and are always equal.
- Same-side interior angles are inside the parallel lines on the same side of the transversal and always add up to $18 0^{\circ}$ (supplementary).

Example: Supplementary angles

Two same-side interior angles formed by a transversal cutting two parallel lines measure $3 x$ and $2 x + 20$ . Because same-side interior angles are supplementary, their measures add to $18 0^{\circ}$ .

Set up the equation:
$3 x + (2 x + 20) = 180$

Combine like terms:
$5 x + 20 = 180$

Subtract $20$ :
$5 x = 160$

Divide by $5$ :
$x = 32$

The angles are $3 (32) = 9 6^{\circ}$ and $2 (32) + 20 = 8 4^{\circ}$ .

Answer: $x = 32$ ; the two angles are $9 6^{\circ}$ and $8 4^{\circ}$

Congruent triangles

You could check all six parts, but it’s usually faster to use one of the standard congruence tests:

SSS: All three sides are equal.
SAS: Two sides and the angle between them are equal.
ASA: Two angles and the included side (the side between those two angles) are equal.
AAS: Two angles and a non-included side (a side not between the two given angles) are equal.
HL: The hypotenuse and one leg in a right triangle are equal.

Common pitfalls

ASA vs AAS: The difference is which side you know. In ASA the known side is between the two given angles; in AAS it is not between them. Mixing these up is one of the most frequent errors on this topic.
Corresponding letters matter in similar figures. When figures are named (e.g., pentagon $A BC D E \sim$ pentagon $PQRST$ ), the letter order tells you which sides correspond. Writing $\frac{A B}{QR}$ instead of $\frac{A B}{PQ}$ is a common proportion error.

Below are examples of two congruence configurations.

Example: Determining congruence (AAS)

Given triangle $A BC$ and triangle $D EF$ with the following:

$∠ A = 2 5^{\circ}$ , $∠ C = 8 3^{\circ}$ , and $A B = 3$ in triangle $A BC$

$∠ D = 2 5^{\circ}$ , $∠ F = 8 3^{\circ}$ , and $D E = 3$ in triangle $D EF$

Are the triangles congruent? If so, state why.

We have two pairs of equal angles: $∠ A = ∠ D = 2 5^{\circ}$ and $∠ C = ∠ F = 8 3^{\circ}$ . The known side $A B = D E = 3$ connects vertices $A$ and $B$ . Because $A B$ does not touch $∠ C$ (or $∠ F$ ), it is not between the two given angles - this is the non-included side condition for AAS.

Therefore $△ A BC ≅ △ D EF$ by AAS, which means all corresponding sides are equal: $BC = EF$ and $C A = F D$ .

Answer: Yes, congruent by AAS

Similar figures and scaling

Two figures are similar if they have the same shape but different sizes. In similar figures:

All corresponding angles are equal.
All corresponding side lengths are proportional by a constant scale factor, denoted $k$ .

When figures are similar, here’s how measurements change:

Angles remain unchanged.
Side lengths scale by a factor of $k$ .
Perimeter scales by $k$ , since each linear measurement is multiplied by $k$ .
Area scales by $k^{2}$ , because area depends on two dimensions.
Volume scales by $k^{3}$ , because volume depends on three dimensions.

Example: Solving for a side in similar polygons

Pentagon $A BC D E$ is similar to pentagon $PQRST$ . If $A B = 6.4 cm$ , $PQ = 10 cm$ , and $BC = 8.1 cm$ , find the length of $QR$ .

Since the pentagons are similar, their corresponding sides are in proportion. $A B$ corresponds to $PQ$ and $BC$ corresponds to $QR$ , so:

$\frac{A B}{PQ} = \frac{BC}{QR} \Rightarrow \frac{6.4}{10} = \frac{8.1}{QR}$

Solve the proportion:

Cross-multiply: $6.4 \cdot QR = 10 \cdot 8.1$

Multiply: $6.4 QR = 81$

Divide: $QR = \frac{81}{6.4} \approx 12.66$

Answer: $12.66 cm$

Example: Scale factor with area and volume

A square with side $4$ has area $16$ . Scaling each side by $k = 3$ gives side $12$ and area $144$ . The ratio $\frac{144}{16} = 9 = 3^{2}$ confirms that area scales by $k^{2}$ . Likewise, scaling each dimension of a solid by $k = 3$ multiplies volume by $3^{3} = 27$ .

Answer: Area scales by $k^{2}$ ( $9 \times$ ); volume scales by $k^{3}$ ( $27 \times$ )

Achievable Praxis Core: Math (5733)

3. Algebra and geometry

Our Praxis Core: Math course is currently in development and is a work-in-progress.

Understanding angles, congruence, and similarity

9 min read

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Discuss

Feedback

Definitions

Vertical angles: Angles opposite each other when two lines intersect; always congruent.
Complementary angles: Two angles whose measures add to $9 0^{\circ}$ .
Supplementary angles: Two angles whose measures add to $18 0^{\circ}$ .
Similar figures: Figures that have the same shape but different sizes, with all corresponding angles equal and side lengths proportional.
Scale factor $k$: The ratio of corresponding side lengths between two similar figures.

Angle relationships

Angles formed by parallel lines and a transversal:
- Corresponding angles are in the same position on each parallel line and are always equal.
- Alternate interior angles are inside the parallel lines on opposite sides of the transversal and are always equal.
- Alternate exterior angles are outside the parallel lines on opposite sides of the transversal and are always equal.
- Same-side interior angles are inside the parallel lines on the same side of the transversal and always add up to $18 0^{\circ}$ (supplementary).

Example: Supplementary angles

Two same-side interior angles formed by a transversal cutting two parallel lines measure $3 x$ and $2 x + 20$ . Because same-side interior angles are supplementary, their measures add to $18 0^{\circ}$ .

Transversal angle measurements

Set up the equation:
$3 x + (2 x + 20) = 180$

Combine like terms:
$5 x + 20 = 180$

Subtract $20$ :
$5 x = 160$

Divide by $5$ :
$x = 32$

The angles are $3 (32) = 9 6^{\circ}$ and $2 (32) + 20 = 8 4^{\circ}$ .

Answer: $x = 32$ ; the two angles are $9 6^{\circ}$ and $8 4^{\circ}$

Congruent triangles

You could check all six parts, but it’s usually faster to use one of the standard congruence tests:

SSS: All three sides are equal.
SAS: Two sides and the angle between them are equal.
ASA: Two angles and the included side (the side between those two angles) are equal.
AAS: Two angles and a non-included side (a side not between the two given angles) are equal.
HL: The hypotenuse and one leg in a right triangle are equal.

Common pitfalls

ASA vs AAS: The difference is which side you know. In ASA the known side is between the two given angles; in AAS it is not between them. Mixing these up is one of the most frequent errors on this topic.
Corresponding letters matter in similar figures. When figures are named (e.g., pentagon $A BC D E \sim$ pentagon $PQRST$ ), the letter order tells you which sides correspond. Writing $\frac{A B}{QR}$ instead of $\frac{A B}{PQ}$ is a common proportion error.

Below are examples of two congruence configurations.

Example: Determining congruence (AAS)

Given triangle $A BC$ and triangle $D EF$ with the following:

$∠ A = 2 5^{\circ}$ , $∠ C = 8 3^{\circ}$ , and $A B = 3$ in triangle $A BC$

$∠ D = 2 5^{\circ}$ , $∠ F = 8 3^{\circ}$ , and $D E = 3$ in triangle $D EF$

Are the triangles congruent? If so, state why.

Triangle ABC

Triangle DEF

We have two pairs of equal angles: $∠ A = ∠ D = 2 5^{\circ}$ and $∠ C = ∠ F = 8 3^{\circ}$ . The known side $A B = D E = 3$ connects vertices $A$ and $B$ . Because $A B$ does not touch $∠ C$ (or $∠ F$ ), it is not between the two given angles - this is the non-included side condition for AAS.

Therefore $△ A BC ≅ △ D EF$ by AAS, which means all corresponding sides are equal: $BC = EF$ and $C A = F D$ .

Answer: Yes, congruent by AAS

Similar figures and scaling

Two figures are similar if they have the same shape but different sizes. In similar figures:

All corresponding angles are equal.
All corresponding side lengths are proportional by a constant scale factor, denoted $k$ .

When figures are similar, here’s how measurements change:

Angles remain unchanged.
Side lengths scale by a factor of $k$ .
Perimeter scales by $k$ , since each linear measurement is multiplied by $k$ .
Area scales by $k^{2}$ , because area depends on two dimensions.
Volume scales by $k^{3}$ , because volume depends on three dimensions.

Example: Solving for a side in similar polygons

Pentagon $A BC D E$ is similar to pentagon $PQRST$ . If $A B = 6.4 cm$ , $PQ = 10 cm$ , and $BC = 8.1 cm$ , find the length of $QR$ .

Since the pentagons are similar, their corresponding sides are in proportion. $A B$ corresponds to $PQ$ and $BC$ corresponds to $QR$ , so:

$\frac{A B}{PQ} = \frac{BC}{QR} \Rightarrow \frac{6.4}{10} = \frac{8.1}{QR}$

Solve the proportion:

Cross-multiply: $6.4 \cdot QR = 10 \cdot 8.1$

Multiply: $6.4 QR = 81$

Divide: $QR = \frac{81}{6.4} \approx 12.66$

Answer: $12.66 cm$

Example: Scale factor with area and volume

A square with side $4$ has area $16$ . Scaling each side by $k = 3$ gives side $12$ and area $144$ . The ratio $\frac{144}{16} = 9 = 3^{2}$ confirms that area scales by $k^{2}$ . Likewise, scaling each dimension of a solid by $k = 3$ multiplies volume by $3^{3} = 27$ .

Square

Square scaled by 3

Answer: Area scales by $k^{2}$ ( $9 \times$ ); volume scales by $k^{3}$ ( $27 \times$ )

Vertical angles are congruent; linear pairs are supplementary.
Complementary angles sum to $9 0^{\circ}$ ; supplementary angles sum to $18 0^{\circ}$ .
Parallel lines cut by a transversal form congruent corresponding, alternate interior, and alternate exterior angles; same-side interior angles are supplementary.
Prove triangle congruence using SSS, SAS, ASA, AAS, or HL (SSA is invalid).
Similar figures have equal angles and proportional sides.
For scale factor $k$ , perimeter scales by $k$ , area by $k^{2}$ , and volume by $k^{3}$ .
Match corresponding parts using the letter order in figure names to solve similarity problems without diagrams.

Understanding angles, congruence, and similarity

Definitions

Vertical angles: Angles opposite each other when two lines intersect; always congruent.
Complementary angles: Two angles whose measures add to $9 0^{\circ}$ .
Supplementary angles: Two angles whose measures add to $18 0^{\circ}$ .
Similar figures: Figures that have the same shape but different sizes, with all corresponding angles equal and side lengths proportional.
Scale factor $k$: The ratio of corresponding side lengths between two similar figures.

Angle relationships

Angles formed by parallel lines and a transversal:
- Corresponding angles are in the same position on each parallel line and are always equal.
- Alternate interior angles are inside the parallel lines on opposite sides of the transversal and are always equal.
- Alternate exterior angles are outside the parallel lines on opposite sides of the transversal and are always equal.
- Same-side interior angles are inside the parallel lines on the same side of the transversal and always add up to $18 0^{\circ}$ (supplementary).

Example: Supplementary angles

Two same-side interior angles formed by a transversal cutting two parallel lines measure $3 x$ and $2 x + 20$ . Because same-side interior angles are supplementary, their measures add to $18 0^{\circ}$ .

Set up the equation:
$3 x + (2 x + 20) = 180$

Combine like terms:
$5 x + 20 = 180$

Subtract $20$ :
$5 x = 160$

Divide by $5$ :
$x = 32$

The angles are $3 (32) = 9 6^{\circ}$ and $2 (32) + 20 = 8 4^{\circ}$ .

Answer: $x = 32$ ; the two angles are $9 6^{\circ}$ and $8 4^{\circ}$

Congruent triangles

You could check all six parts, but it’s usually faster to use one of the standard congruence tests:

SSS: All three sides are equal.
SAS: Two sides and the angle between them are equal.
ASA: Two angles and the included side (the side between those two angles) are equal.
AAS: Two angles and a non-included side (a side not between the two given angles) are equal.
HL: The hypotenuse and one leg in a right triangle are equal.

Common pitfalls

ASA vs AAS: The difference is which side you know. In ASA the known side is between the two given angles; in AAS it is not between them. Mixing these up is one of the most frequent errors on this topic.
Corresponding letters matter in similar figures. When figures are named (e.g., pentagon $A BC D E \sim$ pentagon $PQRST$ ), the letter order tells you which sides correspond. Writing $\frac{A B}{QR}$ instead of $\frac{A B}{PQ}$ is a common proportion error.

Below are examples of two congruence configurations.

Example: Determining congruence (AAS)

Given triangle $A BC$ and triangle $D EF$ with the following:

$∠ A = 2 5^{\circ}$ , $∠ C = 8 3^{\circ}$ , and $A B = 3$ in triangle $A BC$

$∠ D = 2 5^{\circ}$ , $∠ F = 8 3^{\circ}$ , and $D E = 3$ in triangle $D EF$

Are the triangles congruent? If so, state why.

We have two pairs of equal angles: $∠ A = ∠ D = 2 5^{\circ}$ and $∠ C = ∠ F = 8 3^{\circ}$ . The known side $A B = D E = 3$ connects vertices $A$ and $B$ . Because $A B$ does not touch $∠ C$ (or $∠ F$ ), it is not between the two given angles - this is the non-included side condition for AAS.

Therefore $△ A BC ≅ △ D EF$ by AAS, which means all corresponding sides are equal: $BC = EF$ and $C A = F D$ .

Answer: Yes, congruent by AAS

Similar figures and scaling

Two figures are similar if they have the same shape but different sizes. In similar figures:

All corresponding angles are equal.
All corresponding side lengths are proportional by a constant scale factor, denoted $k$ .

When figures are similar, here’s how measurements change:

Angles remain unchanged.
Side lengths scale by a factor of $k$ .
Perimeter scales by $k$ , since each linear measurement is multiplied by $k$ .
Area scales by $k^{2}$ , because area depends on two dimensions.
Volume scales by $k^{3}$ , because volume depends on three dimensions.

Example: Solving for a side in similar polygons

Pentagon $A BC D E$ is similar to pentagon $PQRST$ . If $A B = 6.4 cm$ , $PQ = 10 cm$ , and $BC = 8.1 cm$ , find the length of $QR$ .

Since the pentagons are similar, their corresponding sides are in proportion. $A B$ corresponds to $PQ$ and $BC$ corresponds to $QR$ , so:

$\frac{A B}{PQ} = \frac{BC}{QR} \Rightarrow \frac{6.4}{10} = \frac{8.1}{QR}$

Solve the proportion:

Cross-multiply: $6.4 \cdot QR = 10 \cdot 8.1$

Multiply: $6.4 QR = 81$

Divide: $QR = \frac{81}{6.4} \approx 12.66$

Answer: $12.66 cm$

Example: Scale factor with area and volume

A square with side $4$ has area $16$ . Scaling each side by $k = 3$ gives side $12$ and area $144$ . The ratio $\frac{144}{16} = 9 = 3^{2}$ confirms that area scales by $k^{2}$ . Likewise, scaling each dimension of a solid by $k = 3$ multiplies volume by $3^{3} = 27$ .

Answer: Area scales by $k^{2}$ ( $9 \times$ ); volume scales by $k^{3}$ ( $27 \times$ )