## Math 7 Chapter 2 Lesson 3: The first congruence case of a side – side – side triangle (ccc)

## 1. Summary of theory

### 1.1. Pay attention to draw a triangle with three sides

To draw \(\Delta ABC\) when three sides are known, the length of each side must be less than the sum of the lengths of the other two sides.

### 1.2. Equal Case: Edge-Side-Edge

If the three sides of one triangle are congruent to the three sides of another triangle, then the two triangles are congruent.

If \(\Delta ABC\) and \(\Delta A’B’C’\) have:

\(\begin{array}{l}AB = A’B’\\AC = A’C’\\BC = B’C’\end{array}\)

Then \(\Delta ABC = \Delta A’B’C’\,\,(ccc)\)

## 2. Illustrated exercise

**Question 1: **Given two triangles ABC and ABD with AB=BC=CA=4cm, AD=BD=2cm (and D is on the opposite side of AB).

Prove that \(\widehat {CAD} = \widehat {CBD}\)

**Solution guide**

\(\Delta CAD\) and \(\Delta CBD\) have

AB common side

AC = BC (gt)

AD = BD (gt)

Therefore \(\Delta CAD = \Delta CBD\,\,(ccc)\)

Derive \(\widehat {CAD} = \widehat {CBD}\) (two corresponding angles)

**Verse 2:** For the side drawing. Find a mistake in the following assignment by a student.

\(\Delta {\rm{EFG = }}\Delta {\rm{HGF}}\,\,{\rm{(c}}{\rm{.c}}{\rm{.c)}} \)

Derive \(\widehat {{F_1}} = \widehat {{F_2}}\) (corresponding angle)

So FG is the bisector of angle EFH .

**Solution guide**

In the student’s work, the following inference is false:

\(\Delta {\rm{EFG = }}\Delta {\rm{HGF}}\,\,(ccc)\)

So \(\widehat {{F_1}} = \widehat {{F_2}}\). It is wrong to infer \(\widehat {{F_1}} = \widehat {{F_2}}\) because \(\widehat {{F_1}} = \widehat {{F_2}}\) are not two similar angles. response of the two congruent triangles mentioned above, so FG cannot be inferred as the bisector of \(\widehat {{\rm{EF}}H}.\)

**Question 3: **Let the line segment MN. Draw a central arc. M radius MN and arc with center N radius NM, they intersect at E, F. Prove that:

a) \(\Delta MNE = \Delta MNF\)

b) \(\Delta MEF = \Delta NEF\)

**Solution guide**

a) Consider \(\Delta MNE\) and \(\Delta BNF\) have MN of common edges

ME = MF (=MN, radius)

NE = NF (=NM, radius)

So \(\Delta MNE = \Delta MNF\,\,\,(ccc)\)

b) Consider \(\Delta MEF\) and \(\Delta NEF\) to have a common edge EF

ME = NE (=MN)

MF=NF(=MN)

So \(\Delta MEF = \Delta NEF\,\,(ccc)\)

## 3. Practice

### 3.1. Essay exercises

**Question 1: **Let ABC be a triangle. Draw an arc with center A with radius BC, draw an arc with center C with radius BA, they are spaced between at D (D and B are on opposite sides of AC). Prove that: AD // BC.

**Verse 2: **Triangle ABC has AB = AC. M is the midpoint of BC. Prove that AM is perpendicular to BC.

### 3.2. Multiple choice exercises

**Question 1: **Let the two triangles be congruent. A triangle DEF (no two sides are equal, no two angles are equal) and a triangle with three vertices H,I,K. Knowing that DE = IK, \(\widehat D = \widehat K\). I write

A. \(\Delta ED{\rm{F = }}\Delta KIH\)

B. \(\Delta DE{\rm{F = }}\Delta IKH\)

C. \(\Delta DE{\rm{F = }}\Delta KIH\)

D. \(\Delta DE{\rm{F = }}\Delta WHEN\)

**Verse 2: **Given \(\Delta ABC{\rm{ = }}\Delta D{\rm{EF}}\) there is \(\widehat B = {70^0};\widehat C = {50^0}{\rm {;EF = 3cm}}\). The measure of angle D and the length of side BC is:

A. \(\widehat D = {60^0};BC = 4cm\)

B. \(\widehat D = {60^0};BC = 3cm\)

C. \(\widehat D = {70^0};BC = 3cm\)

D. \(\widehat D = {80^0};BC = 4cm\)

**Question 3: **Given the following figure. Which triangle is equal to triangle ABC?

A. \(\Delta BAC{\rm{ = }}\Delta A{\rm{ED}}\)

B. \(\Delta ACB{\rm{ = }}\Delta A{\rm{ED}}\)

C. \(\Delta ABC{\rm{ = }}\Delta A{\rm{ED}}\)

D. \(\Delta ABC{\rm{ = }}\Delta A{\rm{DE}}\)

**Question 4: **Let two triangles ABD and CDB have common side BD. Know AB = DC and AD = CB. Which of the following statements is false?

A. \(\Delta ABC{\rm{ = }}\Delta C{\rm{DA}}\)

B. \(\widehat {ABC} = \widehat {CDA}\)

C. \(\widehat {BAC} = \widehat {DAC}\)

D. \(\widehat {BCA} = \widehat {DAC}\)

**Question 5: **For the picture below.

Choose the wrong sentence.

A. AD // BC

B. AB // CD

C. \(\Delta ABC{\rm{ = }}\Delta {\rm{CDA}}\)

D. \(\Delta ABC{\rm{ = }}\Delta A{\rm{DC}}\)

## 4. Conclusion

Through this lesson, you should achieve the following goals:

- Know how to draw a triangle when knowing the 3 sides.
- Prove that two triangles are congruent in the side-side-side case.
- Do related math problems.

.

=============