Practice Questions

Define the term 'congruent figures' in geometry.

Critique the following argument: "In $\triangle XYZ$ and $\triangle LMN$, it is given that $XY = LM$, $YZ = MN$, and $\angle X = \angle L$. Since three parts are equal, the triangles must be congruent."

Justify why the AAA (Angle-Angle-Angle) condition is not sufficient for proving the congruence of two triangles. Provide a simple diagrammatic counterexample.

Name the congruence rule used for two right-angled triangles.

List the five main criteria for proving the congruence of triangles.

In quadrilateral $PQRS$, $PS = QR$ and $\angle SPQ = \angle RQP = 90^\circ$. Demonstrate that $\triangle PQS \cong \triangle QPR$.

In $\triangle PQR$, if $PQ = PR$ and $\angle Q = 55^\circ$, calculate the measure of $\angle P$.

If $\triangle ABC \cong \triangle XYZ$, with $AB = 5$ cm, $\angle B = 60^\circ$, and $\angle A = 40^\circ$, determine the length of side $XY$ and the measure of $\angle Z$.

In $\triangle LMN$ and $\triangle GHI$, it is given that $LM = GH$, $MN = HI$, and $LN = GI$. Identify the congruence criterion that can be used to show that $\triangle LMN \cong \triangle GHI$.

What does the abbreviation 'CPCT' stand for in the context of congruent triangles?

State the Side-Angle-Side (SAS) congruence rule.

Formulate a proof for the statement: "If the bisector of an angle of a triangle is also perpendicular to the opposite side, then the triangle is isosceles."

In the given figure, $PS$ is an altitude of an isosceles triangle $PQR$ in which $PQ = PR$. Analyze the triangles $\triangle PQS$ and $\triangle PRS$ to show that $PS$ bisects the base $QR$.

If two triangles have all three of their corresponding angles equal, are they necessarily congruent? Explain your answer in one sentence.

Explain what is meant by 'one-one correspondence' between the vertices when we state that $\triangle ABC \cong \triangle PQR$.

Given that $\triangle XYZ \cong \triangle LMN$, list all the corresponding congruent parts of the two triangles.

Describe an equilateral triangle and state the measure of each of its interior angles.

Analyze two right-angled triangles, $\triangle ABC$ and $\triangle DEF$, where $\angle B = \angle E = 90^\circ$. If hypotenuse $AC = DF$ and side $AB = DE$, which congruence rule demonstrates that $\triangle ABC \cong \triangle DEF$?

In $\triangle ABC$, the bisectors of $\angle B$ and $\angle C$ meet at point I. A line parallel to BC is drawn through I, intersecting AB at D and AC at E. Formulate and prove a relationship between the length of segment DE and the lengths of segments BD and CE.

Describe the two main properties of an isosceles triangle concerning its sides and angles.

Explain the ASA (Angle-Side-Angle) congruence rule in detail. Describe which parts of the two triangles must be equal and clarify the meaning of the 'included side'.

In the figure, lines $l$ and $m$ are parallel, and they are intersected by a transversal $t$. The bisectors of a pair of interior angles on the same side of the transversal intersect at point $P$. Show that $\angle APB = 90^\circ$.

Line segment $AB$ is parallel to another line segment $CD$. $M$ is the midpoint of $AD$. Show that $M$ is also the midpoint of $BC$.

Examine the figure where $O$ is the center of a circle and $OM$ is perpendicular to chord $AB$. If the radius $OA = 5$ cm and chord $AB = 8$ cm, calculate the length of $OM$.

In the given figure, $AC = BC$, $\angle DCA = \angle ECB$ and $\angle DBC = \angle EAC$. Demonstrate that $\triangle DBC \cong \triangle EAC$.

$ABC$ is an isosceles triangle with $AB = AC$. Side $BA$ is produced to a point $D$ such that $AD = AB$. Analyze the angles in the figure and show that $\angle BCD$ is a right angle.

Justify why the first step in Example 7 of the source text is to prove $\triangle PAQ \cong \triangle PBQ$. Why can one not directly prove $\triangle PAC \cong \triangle PBC$ using the initially given information?

Evaluate the statement: "Any two right-angled triangles with the same hypotenuse length are congruent." Justify your conclusion.

Propose a method to prove that the altitudes from the vertices to the equal sides of an isosceles triangle are equal. State the congruence rule you would use.

In a quadrilateral ABCD, it is given that $AB = AD$ and $CB = CD$. Formulate and prove the conjecture that diagonal AC is the perpendicular bisector of diagonal BD.

In a right-angled triangle $\triangle ABC$, with the right angle at B, D is a point on the hypotenuse AC such that BD is perpendicular to AC. Create a proof that shows $\angle ABD = \angle C$.

Formulate a proof for Theorem 7.3: "The sides opposite to equal angles of a triangle are equal," using the ASA congruence rule.

Critique the proof of Theorem 7.2 in the source text, which uses an angle bisector. Propose an alternative proof that uses a median from vertex A to the base BC, and justify each step.

Describe the logical steps used to prove Theorem 7.2, which states that 'Angles opposite to equal sides of an isosceles triangle are equal'. You do not need to write a formal proof, but list the main steps including any construction required.

Summarize the key differences between the SAS, ASA, and SSS congruence rules. For each rule, identify the three corresponding parts that must be shown to be equal.

In a parallelogram $ABCD$, points $P$ and $Q$ are taken on the diagonal $BD$ such that $DP = BQ$. Show that quadrilateral $APCQ$ is a parallelogram.

Let $\triangle ABC$ be an isosceles triangle with $AB = AC$. The side BA is produced to D such that $AD = AB$. Justify that $\angle BCD$ is a right angle.

AD, BE, and CF are three equal altitudes of $\triangle ABC$. Using the RHS congruence rule, demonstrate that $\triangle ABC$ is an equilateral triangle.

In $\triangle ABC$, D is the mid-point of side AC and E is the mid-point of side AB. The medians BD and CE are extended to points P and Q respectively, such that $BD = DP$ and $CE = EQ$. Create a proof to show that Q, A, and P are collinear.

Summarize the relationship between equal sides and equal angles in a triangle, as described by Theorem 7.2 and its converse, Theorem 7.3.

In $\triangle ABC$, $D$ is the mid-point of side $BC$. The perpendiculars from $D$ to $AB$ and $AC$ are $DE$ and $DF$ respectively, and $DE = DF$. Demonstrate that $\triangle ABC$ is an isosceles triangle.

Propose a proof to show that in any triangle, the sum of any two sides is greater than twice the length of the median drawn to the third side.

Design a proof to show that if the medians drawn to two sides of a triangle are equal, then the triangle is isosceles.

In a right-angled triangle $ABC$, right-angled at $B$, $M$ is the mid-point of the hypotenuse $AC$. $B$ is joined to $M$ and produced to a point $D$ such that $BM = MD$. Point $D$ is joined to $C$. Show that $BM = \frac{1}{2} AC$.

Explain why the SSA (Side-Side-Angle) condition is not accepted as a valid rule for triangle congruence.