## Math 8 Chapter 1 Lesson 1: Quadrilaterals

## 1. Theoretical Summary

### 1.1 Quadrilaterals

– Quadrilateral ABCD is a figure consisting of four line segments AB, BC, CD, DA in which any two line segments do not lie on the same line.

A simple quadrilateral is a quadrilateral whose sides only intersect at the vertex.

A convex quadrilateral is a simple quadrilateral that always lies in a semi-plane whose edge is a line containing any side of the quadrilateral.

### 1.2 Properties

**a) Diagonal property**

It is proved that:

- In a convex quadrilateral, two diagonals intersect at a point in the interior of the quadrilateral.
- Conversely, if a quadrilateral has two diagonals intersecting at a point in its interior, then the quadrilateral is a convex quadrilateral.

**b) Angle property**

**Theorem: **The sum of the four angles of a quadrilateral is 360^{o}

**Prove:**

“To prove proposition A is true, we assume that a is false. From the false hypothesis A we draw an absurd conclusion (contrary to the hypothesis or contrary to the theorems, axioms, or contrary to the results). correct argument we have).”

So A is correct.

## 2. Illustrated exercise

**Question 1: **Let quadrilateral ABCD have angles A, B, C, D whose measure is proportional to the numbers 1; 2; 3; 4.

Calculate the measure of the angles \(\widehat A;\widehat B;\widehat C;\widehat D\)

**Solution guide**

We have: \(\frac{{\widehat A}}{1} = \frac{{\widehat B}}{2} = \frac{{\widehat C}}{3} = \frac{{\widehat D}}{4}\)

By the property of the series of equal ratios, we get:

\(\frac{{\widehat {\rm{A}}}}{1} = \frac{{\widehat {\rm{B}}}}{2} = \frac{{\widehat {\rm{ C}}}}{3} = \frac{{\widehat {\rm{D}}}}{4} = \frac{{\widehat {\rm{A}} + \widehat {\rm{B} } + \widehat {\rm{C}} + \widehat {\rm{D}}}}{{1 + 2 + 3 + 4}} = \frac{{{{360}^o}}}{{ 10}} = {36^o}\)

( because \(\widehat {\rm{A}} + \widehat {\rm{B}} + \widehat {\rm{C}} + \widehat {\rm{D}} = {360^ \circ } \)).

So:

\(\frac{{\widehat {\rm{A}}}}{1} = {36^ \circ } \Rightarrow \widehat {\rm{A}} = {36^ \circ }\) ; \(\frac{{\widehat {\rm{B}}}}{2} = {36^ \circ } \Rightarrow \widehat {\rm{B}} = {72^ \circ }\);

\(\frac{{\widehat {\rm{C}}}}{3} = {36^ \circ } \Rightarrow \widehat {\rm{C}} = {108^ \circ }\); \(\frac{{\widehat {\rm{D}}}}{4} = {36^ \circ } \Rightarrow \widehat {\rm{D}} = {144^ \circ }\).

**Verse 2: **Given quadrilateral ABCD, know AB = AD; \(\widehat {\rm{B}} = {90^ \circ };\widehat {\rm{A}} = {60^ \circ }\) and \(\widehat {\rm{D}} = {135^ \circ }\).

a. Calculate the angle \(\widehat {\rm{C}}\) and prove that BD = BC.

b. From A, draw AE perpendicular to the line CD. Calculate the angles of triangle AEC.

**Solution guide**

a. We have:

\(\begin{array}{l} \widehat {\rm{C}} = {360^ \circ } – ({60^ \circ } + {90^ \circ } + {135^ \circ })\\ \Rightarrow \widehat {\rm{C}} = {75^ \circ } \end{array}\)

Triangle ABD has AB = AD and \(\widehat {\rm{A}} = {60^ \circ }\)

so it is an equilateral triangle, deducing:

\(\widehat {{\rm{D}}_1^{}} = {60^ \circ }{\rm{ }}\) and \( \widehat {{\rm{D}}_2^{}} = {135^ \circ }{\rm{ – }}\widehat {{\rm{D}}_1^{}}{\rm{ = 13}}{{\rm{5}}^ \circ }{ \rm{ – }}{60^ \circ } = {75^ \circ }{\rm{ }}\)

The CBD triangle has \(\widehat {\rm{C}} = {\rm{ }}\widehat {{\rm{D}}_2^{}} = {75^ \circ }{\rm{ }} \) so it is an isosceles triangle. So BD = BC.

b. Quadrilateral ABCE has \(\widehat {\rm{B}} = {90^ \circ }{\rm{,}}\widehat {\rm{E}} = {90^ \circ }{\rm{; }}\widehat {\rm{C}}{\rm{ = 7}}{{\rm{5}}^ \circ }{\rm{ }}\) so: \(\widehat {{\rm{ EAB}}}{\rm{ = 36}}{0^ \circ } – ({90^ \circ } + {90^ \circ } + {75^ \circ }) = {105^ \circ }\)

We have: BC = BD where BD = BA \( \Rightarrow \) BC = BA

\( \Rightarrow \) \(\Delta {\rm{ABC}}\) is an isosceles right triangle so: \(\widehat {{\rm{BAC}}} = {45^ \circ }\).

we have: \(\widehat {{\rm{CAE}}} = {105^ \circ } – {45^ \circ } = {60^ \circ } \Rightarrow \widehat {{\rm{ACE}}} = {90^ \circ } – {60^ \circ } = {30^ \circ }\)

Note: you can calculate \(\widehat {{\rm{ACE}}} \) first;

\(\Delta {\rm{ABC}}\) square \( \Rightarrow \widehat {{\rm{BCA}}} = {45^ \circ }.\)

\(\widehat {{\rm{EAC}}} = \widehat {{\rm{ECB}}} – \widehat {{\rm{ACB}}} = {75^ \circ } – {45^ \circ } = {30^ \circ }\).

### 3.1. Essay exercises

**Question 1: **Let quadrilateral ABCD have angles A, B, C, D whose measure is proportional to the numbers 1; 2; 4; 5. Calculate the measure of the angles \(\widehat A;\widehat B;\widehat C;\widehat D\)

**Verse 2:** Given quadrilateral ABCD as shown, there are \(AB=AD;\,\,CB=CD.\)

a) Prove that AC is the perpendicular bisector of BD.

b) Calculate the angles \(\widehat{B}\,\,v\grave{a}\,\,\widehat{D}\) of a quadrilateral known \(\widehat{A}=90{}^\ circ ;\,\,\widehat{C}=45{}^\circ \)

### 3.2. Multiple choice exercises

**Question 1: **The sum of the 4 angles of a quadrilateral is equal to:

A. \(180^o\)

B. \(90^o\)

C. \(360^o\)

D. \(540^o\)

**Verse 2: **Let quadrilateral ABCD know \(\angle A + \angle B = {160^0}\) ask \(\angle C + \angle D =?\)

A. \({200^0}\)

B. \({20^0}\)

C. \({260^0}\)

D. \({320^0}\)

**Question 3: **Let ABC be a quadrilateral whose angles are \(\angle A = {60^0}\,\angle B = {140^0}\,\,\angle C = {30^0}\) \(\angle D = ?\)

A. \({20^0}\)

B. \({120^0}\)

C. \({130^0}\)

D. \({150^0}\)

**Question 4: **Let ABCD know that \(\angle A = {80^0}\,\angle B = {110^0}\,\,\angle C = {40^0}\) asks the measure of the exterior angle at vertex D how much?

A. \({150^0}\)

B. \({130^0}\)

C. \({120^0}\)

D. \({50^0}\)

**Question 5: **Let quadrilateral ABCD know \(\angle B = {50^0}\) and angle A is twice the angle B and angle C is twice the angle D. What is the measure of the angles of quadrilateral ABCD?

A. \(\angle A = {100^0}\,\,\,\angle B = {50^0}\,\,\,\,\angle C = {140^0}\,\,\ angle D = {70^0}\)

B. \(\angle A = {90^0}\,\,\,\angle B = {60^0}\,\,\,\,\angle C = {140^0}\,\,\ angle D = {70^0}\)

C. \(\angle A = {80^0}\,\,\,\angle B = {70^0}\,\,\,\,\angle C = {140^0}\,\,\ angle D = {70^0}\)

D. \(\angle A = {80^0}\,\,\,\angle B = {50^0}\,\,\,\,\angle C = {160^0}\,\,\ angle D = {70^0}\)

Through this lesson, you should achieve the following goals:

- Identify quadrilaterals.
- Remember the properties of quadrilaterals.
- Apply knowledge to solve some related problems.

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