Quick Review Flashcards - Click to flip and test your knowledge!
Question
What is the definition of an isosceles triangle?
Answer
A triangle with at least two sides equal to each other.
Question
How is an equilateral triangle defined?
Answer
A triangle in which all the sides are equal to each other.
Question
What is the relationship between the properties of isosceles and equilateral triangles?
Answer
An equilateral triangle satisfies all properties of an isosceles triangle, but an isosceles triangle does not necessarily satisfy those of an equilateral triangle.
Question
According to Theorem 1, what is true of the angles opposite to the equal sides of a triangle?
Answer
The angles opposite to equal sides are also equal.
Question
In Theorem 1, which congruence criterion is used to prove that $\angle B = \angle C$?
Answer
The $R.H.S.$ (Right angle-Hypotenuse-Side) criterion.
Question
According to Theorem 2, what is true of the sides opposite to equal angles in a triangle?
Answer
The sides opposite to equal angles are also equal.
Question
In Theorem 2, which congruence criterion is used to prove that $AB = AC$?
Answer
The $A.A.S.$ (Angle-Angle-Side) criterion.
Question
What effect does the bisector of the vertex angle of an isosceles triangle have on the base?
Answer
It bisects the base at right angles.
Question
Where does the perpendicular bisector of the base of an isosceles triangle always pass through?
Answer
It passes through the vertex of the triangle.
Question
What is the property of the exterior angles formed when the equal sides of an isosceles triangle are produced?
Answer
The exterior angles so formed are equal to each other.
Question
In $\triangle ABC$, if $AB = BC$ and $AC = CD$ (where $D$ is on $BC$ produced), what is the ratio of $\angle BAD$ to $\angle ADB$?
Answer
The ratio is $3 : 1$.
Question
In an isosceles triangle, what is the relationship between the line joining the mid-point of the base to the opposite vertex and the base itself?
Answer
The line is perpendicular to the base.
Question
If $D$ and $E$ are the mid-points of equal sides $AB$ and $AC$ in $\triangle ABC$, what is the relationship between the segments $BE$ and $CD$?
Answer
The segments $BE$ and $CD$ are equal.
Question
In $\triangle ABC$, if $AB = AC$ and $BO$ and $CO$ bisect the base angles, what does the line $AO$ do to the vertex angle?
Answer
The line $AO$ bisects the angle $\angle BAC$.
Question
In $\triangle ABC$ where $AB = AC$, if a line $DE \parallel BC$ passes through an interior point $P$ formed by base angle bisectors, how is $DE$ calculated?
Answer
$DE = BD + CE$.
Question
Under what condition is the bisector of an exterior angle at the vertex of a triangle parallel to the base?
Answer
The condition is that the triangle must be an isosceles triangle.
Question
What does the abbreviation CPCTC stand for in geometric proofs?
Answer
Corresponding parts of congruent triangles are congruent.
Question
In the context of triangle inequalities, what does the symbol $>$ represent?
Answer
It represents the phrase "is greater than".
Question
According to Theorem 3, if two sides of a triangle are unequal, which side has the greater angle opposite to it?
Answer
The longer side has the greater angle opposite to it.
Question
What is the relationship between an exterior angle of a triangle and its interior opposite angles?
Answer
The exterior angle is always greater than each of its interior opposite angles.
Question
According to Theorem 4, if two angles of a triangle are unequal, which angle has the longer side opposite to it?
Answer
The greater angle has the longer side opposite to it.
Question
In any right-angled triangle, why is the hypotenuse always the longest side?
Answer
The hypotenuse is opposite the right angle, which is the greatest angle in the triangle.
Question
According to Theorem 5, which line segment from a point to a straight line represents the shortest distance?
Answer
The perpendicular line segment is the shortest.
Question
What does Corollary 1 state regarding the sum of the lengths of any two sides of a triangle?
Answer
The sum of the lengths of any two sides is always greater than the third side.
Question
What does Corollary 2 state regarding the difference between the lengths of any two sides of a triangle?
Answer
The difference between the lengths of any two sides is always less than the third side.
Question
In $\triangle ABC$, express the Triangle Inequality Theorem for side $BC$ using the other two sides.
Answer
$BC < AB + AC$.
Question
In $\triangle ABC$, express the relationship between the difference of sides $AB$ and $AC$ and the third side $BC$.
Answer
$AB - AC < BC$.
Question
If $AD$ is a median of $\triangle ABC$, what is the relationship between the sum of sides $AB + AC$ and the median?
Answer
$AB + AC > 2AD$.
Question
How does the perimeter of a triangle compare to the sum of the lengths of its medians?
Answer
The perimeter of a triangle is greater than the sum of the lengths of its medians.
Question
For any point $P$ in the interior of $\triangle ABC$, what is the inequality relating $PA + PB$ to the sides of the triangle?
Answer
$PA + PB < AC + BC$.
Question
If $AD$ bisects $\angle BAC$ in $\triangle ABC$, what is the relationship between side $AB$ and segment $BD$?
Answer
$AB > BD$.
Question
In quadrilateral $ABCD$, if $AB$ is the shortest side and $CD$ is the longest side, what is the relationship between $\angle B$ and $\angle D$?
Answer
$\angle B > \angle D$.
Question
In quadrilateral $ABCD$, if $AB$ is the shortest side and $CD$ is the longest side, what is the relationship between $\angle A$ and $\angle C$?
Answer
$\angle A > \angle C$.
Question
If $\angle B = \angle C$ and $\angle BAD = \angle CAD$ in $\triangle ABC$, what is the resulting relationship between $AB$ and $AC$?
Answer
$AB = AC$.
Question
If $AD \perp BC$ and $\triangle ABD \cong \triangle ACD$, what is the relationship between segments $BD$ and $CD$?
Answer
$BD = CD$.
Question
In an isosceles $\triangle ABC$ where $AB = AC$, what is the value of the exterior angle at $C$ if $\angle B = 50^\circ$?
Answer
The exterior angle is $130^\circ$.
Question
If in $\triangle ABC$, $AB = AC$ and $\angle A = 100^\circ$, what is the measure of $\angle B$?
Answer
$\angle B = 40^\circ$.
Question
In $\triangle ABC$, if $AB = AC$, $BO$ bisects $\angle B$, and $CO$ bisects $\angle C$, why does $OB = OC$?
Answer
Because the base angles are equal, their halves are also equal, making $\triangle BOC$ isosceles.
Question
In the proof of Theorem 3, why is $\angle ADC > \angle B$?
Answer
Because $\angle ADC$ is an exterior angle to $\triangle BDC$.
Question
If $\angle BAC = 80^\circ$ and $AD$ bisects $\angle BAC$ in $\triangle ABC$, what are the measures of $\angle BAD$ and $\angle CAD$?
Answer
Both angles measure $40^\circ$.
Question
In $\triangle ABC$ where $AB = AC$, if $\angle B = 60^\circ$, what specific type of triangle is it?
Answer
It is an equilateral triangle.
Question
Which side is the longest in a triangle with angles $40^\circ$, $60^\circ$, and $80^\circ$?
Answer
The side opposite the $80^\circ$ angle.
Question
The property that "sum of any two sides of a triangle is always greater than the third side" is often called the _____.
Answer
Triangle Inequality Theorem
Question
In $\triangle ABC$, if $AB = 7$ cm and $BC = 10$ cm, what is the minimum possible whole number length for side $AC$?
Answer
$4$ cm (since $AC > 10 - 7$).
Question
In $\triangle ABC$, if $AB = 7$ cm and $BC = 10$ cm, what is the maximum possible whole number length for side $AC$?
Answer
$16$ cm (since $AC < 10 + 7$).
Question
In $\triangle ABC$, if $\angle A = 90^\circ$ and $P$ is a point on $AC$, why is $PB > AB$?
Answer
In right-angled $\triangle ABP$, $PB$ is the hypotenuse and therefore the longest side.
Question
If $AD$ is a median of $\triangle ABC$, to what point $E$ is $AD$ produced to prove $AB + AC > 2AD$?
Answer
It is produced such that $AD = DE$.
Question
How does the measure of a vertex angle of an isosceles triangle relate to its base angles?
Answer
The vertex angle is equal to $180^\circ$ minus twice the measure of one base angle.
Question
In Theorem 2, which side is identified as "Common" to both $\triangle ABD$ and $\triangle ACD$?
Answer
The side $AD$.
Question
If the three medians of a triangle are $m_1$, $m_2$, and $m_3$, and the perimeter is $P$, write the inequality.
Answer
$P > m_1 + m_2 + m_3$.
Question
In $\triangle ABC$, if $AB > AC$, how does the median $AD$ to the base $BC$ relate to the angles $\angle ADB$ and $\angle ADC$?
Answer
$\angle ADB$ is an obtuse angle while $\angle ADC$ is an acute angle (implying $AB > AC$).
Question
If $D$ is the mid-point of $BC$ in isosceles $\triangle ABC$ ($AB=AC$), what is the measure of $\angle ADB$?
Answer
$90^\circ$.
Question
If an isosceles triangle has a vertex angle of $100^\circ$, what is the measure of each exterior angle at the base?
Answer
$140^\circ$ (since each base angle is $40^\circ$).
Question
In the inequality proofs, what is the reason for stating $AC = AD$ in $\triangle ACD$?
Answer
By construction.
Question
In $\triangle ABC$, if $AB=AC$ and $D$ is any point on $BC$ produced, why is $AD > AB$?
Answer
Because $\angle ABC$ (ext angle to $\triangle ABD$) is greater than $\angle ADB$.
Question
When comparing two triangles $ABC$ and $PQR$, if $AB=AC$, $\angle C=\angle P$, and $\angle B=\angle Q$, the triangles are described as _____.
Answer
isosceles and congruent.
Question
In $\triangle ABC$, if $AB=AC$ and $AD \perp BC$, which line segment represents the altitude of the triangle?
Answer
The segment $AD$.
Question
If the side $BC$ of $\triangle ABC$ is produced to $D$, the exterior angle $\angle ACD$ is equal to the sum of which two angles?
Answer
$\angle CAB$ and $\angle ABC$.
Question
In $\triangle ABC$, if $AB$ is the longest side, which angle is the largest?
Answer
$\angle C$ (the angle opposite to side $AB$).
Question
What is the relationship between $x$ and $y$ if $BD = x$ and $DC = y + 1$ in congruent $\triangle ABD$ and $\triangle ACD$?
Answer
$x = y + 1$.