 # Congruent and Similar Triangles

### Congruent and Similar Triangles

#### Similar Triangles

Similar triangles are two triangles that have the same shape but not identical or not same size.

In 2 similar triangles, the corresponding angles are equal and the corresponding sides have the same ratio.

There are 3 ways of Similarity Tests to prove for similarity between two triangles:

###### 1.     AAA (Angle, Angle, Angle) If two angles are equal (which implies three angles of the two triangles are equal) then the triangles are similar.

###### 2.    Equal ratio of 3 corresponding sides $$\hspace{3em} {\large\frac{a_{1}}{a_{2}}} \ = \ {\large\frac{b_{1}}{b_{2}}} \ = \ {\large\frac{c_{1}}{c_{2}}} \ = \ k$$

###### 3.    Equal ratio of 2 corresponding sides and equal angle in between the two sides $$\\[20pt] \hspace{3em} {\large\frac{a_{1}}{a_{2}}} \ = \ {\large\frac{b_{1}}{b_{2}}} \ = \ k$$
$$\\[20pt] \hspace{2.9em} {\small \angle B_{1} \ = \ \angle B_{2} }$$

#### Congruent Triangles

Congruent triangles are two triangles that have the same shape and identical or same size.

In 2 congruents triangles, the corresponding angles and the corresponding sides are equal.

There are 4 ways of Congruence Tests to prove for congruence between two triangles:

###### 1.     SSS (Side, Side, Side) Each corresponding sides of congruent triangles are equal (side, side, side).

###### 2.     SAS (Side, Angle, Side) Two corresponding sides and the angle in between of two congruent triangles are equal (side, angle, side).

###### 3.     AAS (Angle, Angle, Side) Two angles and another opposide side of two congruent triangles are equal (angle, angle, side).

###### 4.     RHS (Right Angle, Hypotenuse, Side) The hypotenuse and one perpendicular side of two congruent right-angled triangles are equal (right angle, hypotenuse, side).

Try some of the examples below and if you need any help, just look at the solution I have written. Cheers ! =) .
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EXAMPLE:

$${\small 1.\enspace}$$ In the diagram, $${\small P\hat{Q}R}$$ = $${\small Q\hat{R}S}$$ = $${\small R\hat{S}T}$$ = $${\small 90^{\large{\circ}}}$$, $${\small PQ}$$ = $${\small QR}$$ = $${\small RS}$$ = 5 cm and $${\small ST}$$ = 1cm. Find the length of $${\small QU}$$.
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$${\small 2.\enspace}$$ Name the triangle which is similar to triangle PQR and calculate the values of the unknown lengths.
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$${\small\hspace{1.2em}(\textrm{a}).\hspace{0.8em}}$$
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$${\small\hspace{1.2em}(\textrm{b}).\hspace{0.8em}}$$
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$${\small\hspace{1.2em}(\textrm{c}).\hspace{0.8em}}$$
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$${\small\hspace{1.2em}(\textrm{d}).\hspace{0.8em}}$$
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$${\small 3.\enspace}$$ In the diagram, AB = 5 cm, BD = 12 cm and DE = 7 cm. Calculate BC.
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$${\small 4.\enspace}$$ In the diagram shown, PQ, LM and RS are parallel. If PQ = 6 cm, RS = 10 cm, calculate the length of LM.
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$${\small 5.\enspace}$$ Given that $${\small \triangle AED}$$ and $${\small \triangle BEC}$$ are isosceles triangles, prove that $${\small \triangle ABE}$$ and $${\small \triangle EDC}$$ are congruent.
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PRACTICE MORE WITH THESE QUESTIONS BELOW!

$${\small 1.\enspace}$$ In the diagram, $${\small AG \parallel BE \ }$$ and $${\small CG \parallel DE}$$, $${\small AB \ = \ CD \ = \ 10 \ }$$ cm and $${\small BC \ = \ 5 \ }$$ cm.
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$${\small\hspace{1.2em}(\textrm{a}).\hspace{0.8em}}$$ Name a pair of congruent triangles.

$${\small\hspace{1.2em}(\textrm{b}).\hspace{0.8em}}$$ Show that the triangles in (a) are congruent.

$${\small\hspace{1.2em}(\textrm{c}).\hspace{0.8em}}$$ Name one triangle that is similar to $${\small \triangle BCF}$$.

$${\small\hspace{1.2em}(\textrm{d}).\hspace{0.8em}}$$ If $${\small DE \ = \ 6 \ }$$ cm, find If $${\small CF }$$.

$${\small 2. \enspace}$$ In the diagram, ABCD is a quadrilateral. Point X is the intersection of AC and BD. AX = XC and BX = DX.
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$${\small\hspace{1.2em}(\textrm{a}).\hspace{0.8em}}$$ Name a pair of congruent triangles.

$${\small\hspace{1.2em}(\textrm{b}).\hspace{0.8em}}$$ Prove that the pair in (a) is congruent.

$${\small\hspace{1.2em}(\textrm{c}).\hspace{0.8em}}$$ What is the name given to the quadrilateral ABCD? Explain your answer.

$${\small 3. \enspace}$$ Are the triangles below congruent? Explain your answer.

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$${\small 4. \enspace}$$ ABCD is a parallelogram. M is the midpoint of AB and N is the point of intersection betweeen BD and MC.
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$${\small\hspace{1.2em} (\textrm{a}).\hspace{0.8em}}$$ Name a triangle which is similar to $${\small \triangle MNB }$$ and show that they are similar.

$${\small\hspace{1.2em} (\textrm{b}).\hspace{0.8em}}$$ Find $${\small {\large \frac{MB}{DC} } }$$.

$${\small\hspace{1.2em} (\textrm{c}).\hspace{0.8em}}$$ Find area of $${\small \triangle DNC }$$ : area of $${\small \triangle BNM }$$.

$${\small 5. \enspace}$$ In the diagram, PR = QS, QO = PO and $${\small \angle PSQ \ = \ 32^{\large{\circ}}}$$.
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$${\small\hspace{1.2em} (\textrm{a}).\hspace{0.8em}}$$ Show that $${\small \triangle POS }$$ and QOR are congruent, stating your reasons clearly.

$${\small\hspace{1.2em} (\textrm{b}).\hspace{0.8em}}$$ Name another pair of congruent triangles.

$${\small 6. \enspace}$$ In the diagram, AD is parallel to BC. AD = 9 cm, BC = 15 cm, EC = 8 cm and DC = 7 cm.
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$${\small\hspace{1.2em} (\textrm{a}).\hspace{0.8em}}$$ Prove that $${\small \triangle ADE }$$ is similar to $${\small \triangle CBE }$$.

$${\small\hspace{1.2em} (\textrm{b}).\hspace{0.8em}}$$ Calculate the length of AE.

$${\small\hspace{1.2em} (\textrm{c}).\hspace{0.8em}}$$ Calculate the ratio of $${\small {\large \frac{\textrm{Area of} \ \triangle ADE}{\textrm{Area of} \ \triangle ADC} }}$$.

$${\small 7. \enspace}$$ ABCD is a rectangle. M is the midpoint of AB and N is a point on CD. MC and BN meet at X.
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$${\small\hspace{1.2em} (\textrm{a}).\hspace{0.8em}}$$ Prove that $${\small \triangle CXN }$$ and $${\small \triangle MXB }$$ are similar.

$${\small\hspace{1.2em} (\textrm{b}).\hspace{0.8em}}$$ Given that the ratio of the area of $${\small \triangle CXN }$$ and $${\small \triangle MXB }$$ is 9 : 4. Find the following ratios

$${\small\hspace{2.8em}(\textrm{i}).\hspace{0.7em}}$$ CN : MB,

$${\small\hspace{2.8em}(\textrm{ii}).\hspace{0.7em}}$$ CN : ND,

$${\small\hspace{2.8em}(\textrm{iii}).\hspace{0.5em}}$$ area of $${\small \triangle BXC }$$ : area of $${\small \triangle BXM }$$.

$${\small 8. \enspace}$$ In the diagram, ABCD is a rectangle and DEFG is a square. AEB and BCG are straight lines. Show that $${\small \angle ADE \ }$$ is equal to $${\small \angle CDG}$$ .
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As always, if you have any particular questions to discuss, leave it in the comment section below. Cheers =) .