Math 8 Chapter 1 Lesson 3: Isosceles trapezoid
1. Summary of theory
1.1. Define
An isosceles trapezoid is a trapezoid with two equal angles adjacent to a base.
ABCD is an isosceles trapezoid (bottom AB; CD)
\( \Leftrightarrow {\rm{ AB // CD }}\) and \({\rm{\hat C = \hat D}}\)
1.2 Properties
Theorem 1: In an isosceles trapezoid, two sides are equal, ABCD is isosceles trapezoid (bottom AB, CD) => AD = BC
Theorem 2: In an isosceles trapezoid, two diagonals are equal, ABCD is an isosceles trapezoid (bottom AB, CD) => AC = BD
Theorem 3: A trapezoid with two equal diagonals is an isosceles trapezoid. Trapezoid ABCD (bottom AB, CD) has AC = BD => ABCD is an isosceles trapezoid.
Signs of an isosceles trapezoid:
– A trapezoid with two equal angles adjacent to the base is an isosceles trapezoid.
A trapezoid with two equal diagonals is an isosceles trapezoid.
Note:
* An isosceles trapezoid has 2 equal sides, but a trapezoid with 2 equal sides is not necessarily an isosceles trapezoid. For example the figure below:
2. Illustrated exercise
Question 1: Let ABCD be an isosceles trapezoid with \(AB\parallel CD\), AB < CD, H and K are the perpendicular projections of A and B onto CD, respectively. Prove that: DK=HC.
Solution guide:
We have the following figure:
Considering two right triangles ADH and BCK, we have:
AD=BC (two sides of an isosceles trapezoid)
\(\angle ADH = \angle BCK\) (two angles adjacent to a base of an isosceles trapezoid)
\( \Rightarrow \Delta ADH = \Delta BCK\) (hypogonal – acute angle)
⇒DH = CK
DH+HK=CK+HK
⇒DK=CH (something to be proved)
Verse 2: Let ABCD be an isosceles trapezoid with \(AB\parallel CD\) ,AB < CD, let E be the intersection of the two sides, F the intersection of the two diagonals. Prove that EF is the perpendicular bisector of AB.
Solution guide:
We have a drawing:
It is easy to see that EAB is an isosceles triangle at E , we have EA=EB so E lies on the perpendicular bisector of AB.(1)
Considering two triangles ABD and BAC we have:
AB is the common side
AD=BD (side of an isosceles trapezoid)
AC=BD (two diagonals of an isosceles trapezoid)
\( \Rightarrow \Delta ABD = \Delta BAC\) (edgeedgeedge)
⇒\(\angle ABD = \angle BAC\)
⇒ AFD is an isosceles triangle at F
⇒AF=BF, so F is also on the perpendicular bisector of AB(2)
From (1) and (2), is the perpendicular bisector of AB (which must be proved).
Question 3: Isosceles trapezoid ABCD with AB, CD as two bases, AB < CD has \(BD \bot BC\) , BD is the bisector of angle D, knowing BC=6 cm. Calculate the perimeter of the trapezoid.
Solution guide:
We have: \(\angle ADC = \angle BCD\)(isosceles trapezoidal property)
But \(\angle BDC = \frac{1}{2}\angle ADC\) (properties of bisectors)
\( \Rightarrow \angle BDC = \frac{1}{2}\angle BCD\)
Besides that, we also have \(\angle BDC + \angle BCD = {90^0}\)
From there we get \(\begin{array}{l} \angle BDC = {30^0}\\ \angle BCD = {60^0} \end{array}\)
Let E be the mid point of CD, considering triangle BEC we have:
\(BE = EC = \frac{1}{2}CD\) (BE is the median to the hypotenuse so it is half of the hypotenuse)
\(\angle BCD = {60^0}\)
⇒Triangle BEC is an equilateral triangle\(BC = BE = EC = \frac{1}{2}CD\)
⇒CD=2.BC=2.6=12 (cm)
We have:
\(\angle ADB = \angle BDE\) (properties of bisectors)
\(\angle BDE = \angle ABD\) (two staggered interior angles)
⇒\(\angle ADB = \angle ABD\)
⇒ The triangle guard ABD is at A
⇒AB=AD
where AD=BC=6 so AB=6 cm
So the perimeter of trapezoid ABCD is: AB+BC+CD+DA=6+6+12+6=30 (cm)
3.1. Essay exercises
Question 1: Calculate the angles of isosceles trapezoid ABCD (AB // CD), knowing D = 2A.
Verse 2: Given triangle ABC is isosceles at A, bisectors BD, CE ( \(D \in AC;\,\,E \in AB\) ). Prove that BEDC is an isosceles trapezoid whose base is as small as its side.
Question 3: Let ABCD be an isosceles trapezoid, whose base AB is equal to the side AD. Prove that AC is the bisector of angle C.
Question 4: Let ABC be an isosceles triangle at A. On the side AB, AC takes the points M, N such that BM = CN.
a) Prove that quadrilateral BMNC is an isosceles trapezoid.
b) Calculate the angles of quadrilateral BMNC knowing that A = 40^{o}.
Question 5: Let ABC be an isosceles triangle at A. On the opposite ray of AC take point D, on the opposite ray of AB take point E such that AD = AE. Prove that quadrilateral BDEC is an isosceles trapezoid.
3.2. Multiple choice exercises
Question 1: Choose the correct statement
A. An isosceles trapezoid is a trapezoid with two equal adjacent angles
B. An isosceles trapezoid is a trapezoid with two equal angles adjacent to a base
C. An isosceles trapezoid is a trapezoid with two equal sides
D. An isosceles trapezoid is a trapezoid with two equal base sides
Verse 2: Choose the wrong statement
A. In an isosceles trapezoid, the two sides are equal
B. In an isosceles trapezoid the two diagonals are equal
C. In an isosceles trapezoid the two diagonals are perpendicular to each other
D. In an isosceles trapezoid, two angles adjacent to a base are congruent
Question 3: Given an isosceles trapezoid ABCD (AB  CD) as shown. Knowing \(\angle D = {60^0}\), what is the measure of angle B?
A. \(\angle B = {60^0}\)
B. \(\angle B = {110^0}\)
C. \(\angle B = {120^0}\)
D. \(\angle B = {80^0}\)
Question 4: Let ABCD (ABCD) be an isosceles trapezoid with AD = 4cm; \(\angle B = {120^0}\) . Choose the correct idea
A. AD = AB= 4cm
B. \(\angle C = \angle B = {120^0}\)
C. AC = BD = 4cm
D. \(\angle C = \angle D = {60^0}\)
Question 5: Choose the correct statement
A. A trapezoid with one right angle is a right trapezoid
B. A quadrilateral with one right angle is a right trapezoid
C. A trapezoid with two perpendicular diagonals is a right trapezoid
D. A quadrilateral with two perpendicular diagonals is a right trapezoid
4. Conclusion
Through this isosceles trapezoid lesson, students need to complete some of the objectives given by the lesson, such as:

Understand the concept of an isosceles trapezoid.

Identify an isosceles trapezoid.

Remember the properties of a trapezoid.

Apply knowledge to solve some related problems.
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