Exercises from chapter 7 — Notable quadrilaterals

Determine the value of $x$ in the following cases:

Determine the angles of the quadrilateral $ABCD$ in the cases:

Determine the value of $x$ in the cases: a) $PA = PB$; b) $AB = AD$ and $CB = CD$.

If $\overline{AP}$ and $\overline{BP}$ are angle bisectors, determine $x$ in the cases:

If $\overline{AP}$ and $\overline{BP}$ are angle bisectors, determine: a) $\hat{C} + \hat{D}$; b) $\hat{C}$, which exceeds $\hat{D}$ by $10^{\circ}$.

If $\overline{BP}$, $\overline{AP}$, $\overline{CQ}$, and $\overline{DQ}$ are angle bisectors, determine $x + y$.

If $ABCD$ is a trapezoid of bases $\overline{AB}$ and $\overline{CD}$, determine $x$ and $y$.

$ABCD$ is a trapezoid of bases $\overline{AB}$ and $\overline{CD}$. If $\overline{DP}$ and $\overline{CP}$ are angle bisectors, determine $x$ and $B\hat{C}D$.

If the trapezoid $ABCD$ is isosceles of bases $\overline{AB}$ and $\overline{CD}$, determine $\hat{A}$.

If $ABCD$ is a parallelogram with $\hat{A} = 2x$ and $\hat{C} = x + 70^{\circ}$, determine $\hat{B}$.

Since $ABCD$ is a parallelogram, $\overline{AP}$ angle bisector, $AB = 7$ cm and $PC = 3$ cm, determine the perimeter of the parallelogram.

If $ABCD$ is a parallelogram, $AD = 20$ cm, $BQ = 12$ cm and $BP = BQ$, determine the perimeter of this parallelogram.

Calculate the sides of a rectangle whose perimeter measures 40 cm, knowing that the base exceeds the height by 4 cm.

Determine the base and height of a rectangle, knowing that the perimeter is 288 m and that the base exceeds three times the height by 4 m.

Calculate the sides of a parallelogram, knowing that its perimeter measures 84 m and that the sum of the smaller sides represents $\dfrac{2}{5}$ of the sum of the longer sides.

The sum of two opposite angles of a parallelogram is equal to $\dfrac{5}{13}$ of the sum of the other two opposite angles. Determine them.

Determine the measures of the angles of a parallelogram, knowing that the difference between two consecutive ones is equal to $\dfrac{1}{9}$ of the sum of their angles.

Prove that the angle bisectors of two consecutive angles of a parallelogram cut at right angles.

In a rectangular trapezoid, the angle bisector of a right angle forms an angle of $110^{\circ}$ with the angle bisector of the acute angle of the trapezoid. Determine the greatest angle of the trapezoid.

The diagonal of a rhombus forms an angle with one of its sides equal to the third of a straight line. Determine the four angles of the rhombus.

The angle bisector of an obtuse angle of the rhombus makes one of the sides an angle of $55^{\circ}$. Determine the value of acute angles.

The largest base of an isosceles trapezoid measures 12 cm and the smallest base 8 cm. Calculate the length of the non-parallel sides, knowing that the perimeter is 40 cm.

One of the interior angles of an isosceles trapezoid is the $\dfrac{2}{7}$ of the adjacent exterior angle. Determine the four angles of the trapezoid.

The sum of two consecutive angles of a trapezoid is equal to $78^{\circ}$ and their difference is $4^{\circ}$. Determine the greatest angle of the trapezoid.

Determine the measures of the angles formed by the internal angle bisectors of a trapezoid in which two consecutive acute angles measure $80^{\circ}$ and $60^{\circ}$.

With a wire 36 m long we build a equilateral triangle and, with the same wire, we then build a square. Determine the ratio of the side of the triangle to the side of the square.

If $ABCD$ is square and $ABP$ is equilateral triangle, determine $x$ in the cases:

Considering the segments with “equal marks” to be congruent, determine the values ​​of the unknowns in the cases:

In the triangle $ABC$ with sides $AB = 9$, $BC = 14$ and $AC = 11$, the points $D$, $E$ and $F$ are midpoints of $\overline{AB}$, $\overline{AC}$ and $\overline{BC}$, respectively. Calculate the perimeter of the triangle $DEF$.

Calculate the perimeter of the triangle $ABC$, with $MN = 7$ cm, $NR = 4$ cm and $MR = 8$ cm, and $M$, $N$, $R$ midpoints of the sides $\overline{AB}$, $\overline{AC}$ and $\overline{BC}$, respectively.

Prove that the midpoints of the sides of any quadrilateral are vertices of a parallelogram.

What conditions must the diagonals of a convex quadrilateral comply with for the midpoints of its sides to be vertices of a rhombus? And a rectangle?

What conditions must the diagonals of a convex quadrilateral comply with for the midpoints of its sides to be vertices of a square?

Let $ABCD$ be a trapezoid with a larger base $\overline{AB}$ and a smaller base $\overline{CD}$, $M$ be the midpoint of the side $\overline{AD}$ and $N$ be the midpoint of $\overline{BC}$. The points $P$ and $Q$ are the intersection points of $\overline{MN}$ with the diagonals $\overline{AC}$ and $\overline{BD}$. Given $AB = a$ and $CD = b$, calculate $MN$, $MP$, $MQ$, $NP$, $NQ$ and $PQ$.

The average base of a trapezoid is 20 cm and the largest base is the $\dfrac{3}{2}$ of the smallest base. Determine the bases.

In a trapezoid, the bases $AB = 20$ cm and $CD = 12$ cm are given. Consider the midpoints $P$ and $Q$ of the diagonals $\overline{AC}$ and $\overline{BD}$ and then the midpoints $R$ and $S$ of the sides $\overline{BC}$ and $\overline{AD}$. Calculate the segments $PR$, $RQ$ and $RS$.

Considering that the segments with “equal marks” are congruent, determine the values ​​of the unknowns in the cases:

In a rectangular trapezoid in which the acute angle measures $45^{\circ}$, the height is equal to the difference between the bases.

Prove that the height of a rectangular trapezoid that has an acute angle measuring $30^{\circ}$ is equal to half of the side not perpendicular to the bases.

Prove that the angle bisectors of the obtuse angles of a parallelogram are parallel.

Prove that the angle bisectors of the angles formed by the diagonals of a rectangle are parallel to the sides of the rectangle.

In an isosceles trapezoid $ABCD$, the smaller base $\overline{AB}$ is congruent to the non-parallel sides. Prove that the diagonals are angle bisectors of the angles $\hat{C}$ and $\hat{D}$ of the trapezoid.

In a parallelogram $ABCD$ we draw its diagonal $\overline{AC}$. Through the vertices $B$ and $D$ we draw two segments $\overline{BP}$ and $\overline{DQ}$ perpendicular to the diagonal $\overline{AC}$, with $P$ and $Q$ in $\overline{AC}$. Prove that $\overline{BP}$ is congruent to $\overline{DQ}$.

Through the midpoint $M$ of the base $\overline{BC}$ of a isosceles triangle $ABC$ we draw the segments $\overline{MP}$ and $\overline{MQ}$ respectively parallel to the sides $\overline{AB}$ and $\overline{AC}$ of the triangle. Prove that $APMQ$ is a rhombus.

Let us consider a convex quadrilateral with two right opposite angles. Prove that the angle bisectors of the other two internal angles of the quadrilateral are rays parallel to each other.

In the figure, $ABCD$ is a square, where $BC + CE = AE$. Since $F$ is the midpoint of $\overline{DC}$, prove that $B\hat{A}E = 2F\hat{A}D$.