## Math 8 Chapter 3 Lesson 7: The third similar case

## 1. Theoretical Summary

**Theorem**

If two sides of one triangle are proportional to two sides of another triangle and the two angles formed by those pairs of sides are congruent, then the two triangles are similar.

## 2. Illustrated exercise

### 2.1. Exercise 1

Which of the triangles below are similar? Please explain (h.41)

**Solution guide**

\(ΔABC\) has \(\widehat A + \widehat B + \widehat C = {180^o}\)

\( \Rightarrow \widehat B + \widehat C = {180^o} – \widehat A\)\(=140^0\)

Where \(ΔABC\) is equal to \(A \Rightarrow \widehat B = \widehat C\)

\(\Rightarrow \widehat B = \widehat C = \dfrac{{140^0}}{2} = {70^o}\)

\(ΔMNP\) at \(P \Rightarrow \widehat M = \widehat N = {70^o}\)

\(ΔABC\) and \(ΔPMN\) have

\(\eqalign{& \widehat B = \widehat M = {70^o} \cr & \widehat C = \widehat N = {70^o} \cr & \Rightarrow \Delta ABC \text{ similarity } \ Delta PMN\,\,\left( {gg} \right) \cr} \)

\(\Delta A’B’C’\) has \(\widehat {A’} + \widehat {B’} + \widehat {C’} = {180^o}\)

\( \Rightarrow \widehat {C’} = {180^o} – \left( {\widehat {A’} + \widehat {B’}} \right) \)\(\,= {180^o} – \left( {{{70}^o} + {{60}^o}} \right) = {50^o}\)

\(ΔA’B’C’\) and \(ΔD’E’F’\) have

\(\eqalign{& \widehat {B’} = \widehat {E’} = {60^o} \cr & \widehat {C’} = \widehat {F’} = {50^o} \cr & \Rightarrow \Delta A’B’C’ \text{ same } \Delta D’E’F’\,\,\left( {gg} \right) \cr} \)

### 2.2. Exercise 2

In Figure 42, AB = 3cm; AC = 4.5cm and ∠(ABD) = ∠(BCA).

a) How many triangles are there in this figure? Are there any pairs of similar triangles?

b) Calculate the lengths x and y (AD = x, DC = y).

c) Add that BD is the bisector of angle B. Calculate the lengths of the lines BC and BD.

**Solution guide**

a) In the figure there are \(3\) triangles: \(ΔABD, ΔCBD, ΔABC\).

\(ΔABD\) and \(ΔACB\) have

\(\widehat B = \widehat C\)

\(\widehat A\) common

\(⇒ ΔABD\) is similar to \(ΔACB\) (gg)

b) \(ΔABD\) is similar to \(ΔACB\)

\(\eqalign{& \Rightarrow {{AB} \over {AD}} = {{AC} \over {AB}} \Rightarrow {3 \over {AD}} = {{4,5} \over 3} \cr & \Rightarrow AD = x = {{3.3} \over {4,5}} = 2 \cr} \)

\(⇒ y = 4.5 – 2 = 2.5\)

c) \(BD\) is the bisector of angle \(B\).

\(\eqalign{ & \Rightarrow {{AB} \over {BC}} = {x \over y} \Rightarrow {3 \over {BC}} = {2 \over {2,5}} \cr & \ Rightarrow BC = {{3.2,5} \over 2} = 3.75 \cr} \)

We have:

\(\eqalign{& \Delta ABD \text{ same }\Delta ACB \cr & \Rightarrow {{AB} \over {BD}} = {{AC} \over {BC}} \Rightarrow {3 \over {BD}} = {{4,5} \over {3.75}} \cr & \Rightarrow BD = {{3.3,75} \over {4,5}} = 2.5 \cr} \)

## 3. Practice

### 3.1. Essay exercises

**Question 1: **Let ABCD be a parallelogram. Let E be the mid point of AB and F the mid point of CD. Prove that triangles ADE and CBF are similar.

**Verse 2: **Right triangle \(ABC\) has \(\widehat A = 90^\circ \) and altitude \(AH.\) From the point \(H\) lower line \(HK\) perpendicular to \(AC) \) (h.27).

a) How many similar triangles are there in the given figure?

b) Write pairs of similar triangles in order of their respective vertices and write the ratio between their respective pairs of sides.

**Question 3: **Trapezoid \(ABCD (AB // CD)\) has \(AB = 2.5cm, AD = 3.5cm,\) \(BD = 5cm\) and \(\widehat {DAB} = \widehat {DBC }\) (h.28).

a) Prove \(∆ ADB\backsim ∆ BCD.\)

b) Calculate the lengths of the sides \(BC, CD\).

c) After calculating, redraw the correct shape with ruler and compass.

**Question 4: **Given a right triangle \(ABC\) (\(\widehat A = 90^\circ \)). Construct \(AD\) perpendicular to \(BC\) (\(D\) belongs to \(BC\)). The bisector \(BE\) intersects \(AD\) at \(F\) (h.29).

Prove: \(\displaystyle {{FD} \over {FA}} = {{EA} \over {EC}}\).

### 3.2. Multiple choice exercises

**Question 1: **Given triangle ABC, AM is the median and has \(\widehat{BAM}=\widehat{BCA}\). Prove it!

A. \(AB^{2}=2BC^{2}\)

B. \(AB^{2}=\sqrt{2}BC^{2}\)

C. \(AB^{2}=\frac{1}{2}BC^{2}\)

D. \(AB^{2}=4BC^{2}\)

**Verse 2: **Choose the best answer. Let ABC be a right triangle at A, M on side AB. Draw \(MD \perp BC (D \in BC)\) MD intersects AC at E

A. Prove that EM.ED=EA.EC

B. Prove that BM.BA=BD.BC

C. Both a and b are correct

D. Both a and b are wrong

**Question 3:** Given triangle ABC, AB=6cm,AC=9cm, \(\widehat{ABD}=\widehat{BCA}\). AD=x. Calculate x

A. 2

B. 3

C. 4

D. 5

**Question 4: **Given trapezoid ABCD (AB//CD), AB=12.5cm,CD=28.5cm, \(\widehat{DAB}=\widehat{DBC}\). So the length BD is closest to which number:

A. 17.5

B. 18

C. 18.5

D. 19

**Question 5:** Choose the correct answer. If two triangles ABC and DEF have \(\widehat{A}=\widehat{D}\), \(\widehat{C}=\widehat{E}\) then:

A. \(\Delta ABC \sim \Delta DEF\)

B. \(\Delta ABC \sim \Delta DFE\)

C. \(\Delta ACB \sim \Delta DFE\)

D. \(\Delta BAC \sim \Delta DFE\)

## 4. Conclusion

Through this lesson, you will learn some of the main topics as follows:

- Understand how to prove that two triangles are similar according to the case gg
- Draw and prove that triangles are similar.

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