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 =) .