📐 geometry

1. **State the problem:** We need to show that the points (4, 4), (3, 5), and (-1, -1) form a right-angled triangle without using the Pythagoras theorem. 2. **Recall the concept:**

1. **Problem statement:** We are given that \(\triangle PQR \cong \triangle TSR\). We know some angles and side lengths, and we need to find the length of \(\overline{PR}\) and the

1. **State the problem:** We need to show that the points (4, 4), (3, 5), and (-1, -1) form a right-angled triangle without using the Pythagoras theorem. 2. **Approach:** Instead o

1. **State the problem:** Prove that triangles $\triangle ABC$ and $\triangle ADC$ are congruent given that $m\angle BAC = m\angle DAC = 51^\circ$, $AB = AD = 4$, and they share si

1. **Problem Statement:** We are given two triangles ABC and BCD sharing side BC. Sides AB and CD are both 5.5 units, sides BD and AC are both 3.4 units. Angles ABC and ACB are 38°

1. **State the problem:** We need to find the coordinates of the image of triangle \(\triangle CDE\) after a 180° counterclockwise rotation about the origin. The original points ar

1. **Problem statement:** Find a point on the x-axis that is equidistant from the points $(7,6)$ and $(3,4)$. 2. **Key idea:** A point on the x-axis has coordinates $(x,0)$. We wan

1. **Problem statement:** Given a circle with points $P$, $V$, and $N$ on the circumference and a point $Q$ outside the circle, we know the lengths $PV = 8$ cm, $NQ = 23$ cm, and w

1. **Problem Statement:** Find the area of a triangular fish pen with sides 16 m, 18 m, and 25 m using Heron's formula. 2. **Formula:** Heron's formula for the area $A$ of a triang

1. Given: □EASY is a parallelogram. 2. In a parallelogram, opposite sides are congruent and opposite angles are congruent.

1. The problem asks to determine the truth value of statements about parallelograms. 2. Statement 1: "A quadrilateral is a parallelogram if both pairs of opposite sides are congrue

1. The problem asks: Which quadrilateral has four right angles and 4 congruent sides? 2. Important definitions:

1. Given that □EASY is a parallelogram, opposite sides are congruent. Therefore, $AS \cong EY$. 2. Opposite angles in a parallelogram are congruent. So, $\angle A \cong \angle S$.

1. Problem: Given parallelogram LOVE with $LO = 16$, find $EV$. Step 1: Recall that in a parallelogram, opposite sides are equal in length.

1. **State the problem:** We need to find the measure of the minor arc $KL$ in a circle where $\angle JNK = 107^\circ$ and $\angle MJN = 99^\circ$. 2. **Understand the setup:** Poi

1. **Problem statement:** We are given a circle with points C, D, E, and F on or around it. Line segment EF is tangent to the circle at point F. We need to find the measure of the

1. **Problem Statement:** Given a star-shaped polygon with 9 outer angles labeled 1 through 9 and 9 inner angles labeled a through i, the sum of the outer angles is given as $$\ang

1. **Problem:** In a circle with center O, the measure of arc AC is 110°. Find the angles \(\angle ADC\) and \(\angle ABC\). 2. **Formula and rules:**

1. **Problem statement:** Calculate the volume of the prism described in 14.c with dimensions 5 m (height), 7 m and 11 m (base sides), and 8 m (depth). 2. **Understanding the shape

1. **Problem Statement:** Given a star with 10 points, the sum of the outer angles labeled \(\angle 1 + \angle 2 + \cdots + \angle 9 = 266^\circ\). Find the sum of the inner angles

1. **Problem Statement:** Find the measure of the angle indicated in bold for problem 21. 2. **Given:** Two intersecting lines form vertical angles. One angle is labeled $x + 55$ (