Sec 4 A-Math · Proofs

Congruent and similar triangles questions with answers

Practice questions on congruent and similar triangles, from the basics up to exam level, each with a full worked solution. Try them on paper first, then open the solution to check your working.

Worked example

$PQ$ is parallel to $SR$ . The lines $PS$ and $QR$ cross at $X$ , and $X$ is the mid-point of $PS$ . Prove that $\triangle PQX \equiv \triangle SRX$ , and state the congruence test.

Step 1 of 2

Match three pairs of parts, each with its reason

\angle QPX = \angle RSX

(alt. ∠s (PQ ∥ SR))

PX = SX

(given)

\angle PXQ = \angle SXR

(vert. opp. ∠s)

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Questions

Question 1Prove two triangles are congruent
The lines $EF$ and $GH$ cross at $K$ , and $K$ is the mid-point of both $EF$ and $GH$ . Prove that $\triangle EKG \equiv \triangle FKH$ , and state the congruence test.
Show the worked solution
1. Step 1: Match three pairs of parts, each with its reason $EK = FK$ (given) $\angle EKG = \angle FKH$ (vert. opp. ∠s) $GK = HK$ (given)
2. Step 2: Name the congruence test the three pairs give $\triangle EKG \equiv \triangle FKH$ (SAS)
Question 2Prove two triangles are congruent
In the diagram, $AB = AD$ and $CB = CD$ . Prove that $\triangle ABC \equiv \triangle ADC$ , and state the congruence test.
Show the worked solution
1. Step 1: Match three pairs of parts, each with its reason $AB = AD$ (given) $CB = CD$ (given) $AC = AC$ (common side)
2. Step 2: Name the congruence test the three pairs give $\triangle ABC \equiv \triangle ADC$ (SSS)
Question 3Prove two triangles are similar, then find a length
In triangle $KLM$ , $N$ lies on $LM$ and $F$ lies on $KL$ such that $\angle LKN = \angle LMF$ . $KL = 6$ cm, $LN = 4$ cm and $NM = 2$ cm. Prove that $\triangle KLN$ is similar to $\triangle MLF$ . Hence find $LF$ .
Show the worked solution
1. Step 1: Two pairs of equal angles, each with its reason $\angle KLN = \angle MLF$ (common ∠) $\angle LKN = \angle LMF$ (given)
2. Step 2: Name the similarity test $\triangle KLN$ is similar to $\triangle MLF$ (AA).
3. Step 3: LM is the whole side: LN + NM $\mathstrut LM = 4 + 2$ $LM = 6$
4. Step 4: Matching sides are in the same ratio: write each side of △MLF over its match $\dfrac{LF}{LN} = \dfrac{ML}{KL}$ $\dfrac{LF}{4} = \dfrac{6}{6}$
5. Step 5: Work out LF $LF = 4$ cm
Question 4Prove two triangles are similar, then find a length
In triangle $ABC$ , $D$ lies on $AB$ and $E$ lies on $AC$ , with $DE$ parallel to $BC$ . $AD = 3$ cm, $DB = 2$ cm and $BC = 6$ cm. Prove that $\triangle ADE$ is similar to $\triangle ABC$ . Hence find $DE$ .
Show the worked solution
1. Step 1: Two pairs of equal angles, each with its reason $\angle ADE = \angle ABC$ (corr. ∠s (DE ∥ BC)) $\angle DAE = \angle BAC$ (common ∠)
2. Step 2: Name the similarity test $\triangle ADE$ is similar to $\triangle ABC$ (AA).
3. Step 3: AB is the whole side: AD + DB $\mathstrut AB = 3 + 2$ $AB = 5$
4. Step 4: Matching sides are in the same ratio: write each side of △ADE over its match $\dfrac{DE}{BC} = \dfrac{AD}{AB}$ $\dfrac{DE}{6} = \dfrac{3}{5}$
5. Step 5: Work out DE $DE = 3.6$ cm

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Worked example

PQ

is parallel to

SR

. The lines

PS

and

QR

cross at

X

, and

X

is the mid-point of

PS

. Prove that

\triangle PQX \equiv \triangle SRX

, and state the congruence test.

Step 1 of 2

Match three pairs of parts, each with its reason

\angle QPX = \angle RSX

(alt. ∠s (PQ ∥ SR))

PX = SX

(given)

\angle PXQ = \angle SXR

(vert. opp. ∠s)

Questions

Question 1Prove two triangles are congruent

The lines $EF$ and $GH$ cross at $K$ , and $K$ is the mid-point of both $EF$ and $GH$ . Prove that $\triangle EKG \equiv \triangle FKH$ , and state the congruence test.

Show the worked solution

Step 1: Match three pairs of parts, each with its reason $EK = FK$ (given) $\angle EKG = \angle FKH$ (vert. opp. ∠s) $GK = HK$ (given)
Step 2: Name the congruence test the three pairs give $\triangle EKG \equiv \triangle FKH$ (SAS)

Question 2Prove two triangles are congruent

In the diagram, $AB = AD$ and $CB = CD$ . Prove that $\triangle ABC \equiv \triangle ADC$ , and state the congruence test.

Show the worked solution

Step 1: Match three pairs of parts, each with its reason $AB = AD$ (given) $CB = CD$ (given) $AC = AC$ (common side)
Step 2: Name the congruence test the three pairs give $\triangle ABC \equiv \triangle ADC$ (SSS)

Question 3Prove two triangles are similar, then find a length

In triangle $KLM$ , $N$ lies on $LM$ and $F$ lies on $KL$ such that $\angle LKN = \angle LMF$ . $KL = 6$ cm, $LN = 4$ cm and $NM = 2$ cm. Prove that $\triangle KLN$ is similar to $\triangle MLF$ . Hence find $LF$ .

Show the worked solution

Step 1: Two pairs of equal angles, each with its reason $\angle KLN = \angle MLF$ (common ∠) $\angle LKN = \angle LMF$ (given)
Step 2: Name the similarity test $\triangle KLN$ is similar to $\triangle MLF$ (AA).
Step 3: LM is the whole side: LN + NM $\mathstrut LM = 4 + 2$ $LM = 6$
Step 4: Matching sides are in the same ratio: write each side of △MLF over its match $\dfrac{LF}{LN} = \dfrac{ML}{KL}$ $\dfrac{LF}{4} = \dfrac{6}{6}$
Step 5: Work out LF $LF = 4$ cm

Question 4Prove two triangles are similar, then find a length

In triangle $ABC$ , $D$ lies on $AB$ and $E$ lies on $AC$ , with $DE$ parallel to $BC$ . $AD = 3$ cm, $DB = 2$ cm and $BC = 6$ cm. Prove that $\triangle ADE$ is similar to $\triangle ABC$ . Hence find $DE$ .

Show the worked solution

Step 1: Two pairs of equal angles, each with its reason $\angle ADE = \angle ABC$ (corr. ∠s (DE ∥ BC)) $\angle DAE = \angle BAC$ (common ∠)
Step 2: Name the similarity test $\triangle ADE$ is similar to $\triangle ABC$ (AA).
Step 3: AB is the whole side: AD + DB $\mathstrut AB = 3 + 2$ $AB = 5$
Step 4: Matching sides are in the same ratio: write each side of △ADE over its match $\dfrac{DE}{BC} = \dfrac{AD}{AB}$ $\dfrac{DE}{6} = \dfrac{3}{5}$
Step 5: Work out DE $DE = 3.6$ cm