ACT 1 Geo: Apply the triangle inequality

Welcome to your ACT 1 Geo: Apply the triangle inequality

$Q_{1}:$ Which sets of numbers could be the lengths of sides of a triangle?

$7,7,7$

$2,3,11$

$1,2,3$

$10,11,22$

$Q_{2}:$ It is possible to have a triangle with all of the following sets of sides EXCEPT

$1,1,1$

$1,2,2$

$3,4,5$

$9,9,18$

$Q_{3}:$ In $\Delta ABC$, $AB = 4$ and $BC = 7$. If the length of the third side is an integer, then what is the greatest possible value for $AC$?

$12$

$14$

$11$

$10$

$Q_{4}:$ For how many integer values of $x$ can $6, 8,$ and $x$ be the lengths of the three sides of a triangle?

$1$

$2$

$10$

$11$

$Q_{5}:$ The lengths of the sides of a triangle are $3, 5,$ and $x$. How many possible values of $x$ are there?

$1$

$5$

$7$

Infinitely many

$Q_{6}:$ The lengths of two sides of a right triangle are $15$ and $17$. Which of the following could be the length of the third side?

$8$

$19$

$\sqrt{514}$

$8, \sqrt{514}$

$Q_{7}:$ If the difference between the measures of the two acute angles of a right triangle is $20$, what is the measure, in degrees, of the smaller one?

$20$

$35$

$45$

$55$

$Q_{8}:$ If one side of $\Delta ABC$ is $5$, what is the smallest integer that could be the perimeter of the triangle?

$8$

$10$

$11$

$12$

$Q_{9}:$ In triangle $ABC$, $AB = BC$. If angle $B$ contains $x$ degrees, find the number of degrees in angle $A$.

$x$

$180-x$

$180-\frac{x}{2}$

$90-\frac{x}{2}$

$Q_{10}:$ If a triangle has side lengths of 7, 7, and 13, what type of triangle is it?

Isosceles triangle

Right triangle

Obtuse-angled triangle

No triangle

$Q_{11}:$ If $ a, b,$ and $c$ are the side lengths of the triangle, then one of the following inequality is true?

$a < c < b$

$a+ b < c < a - b$

$a - b < c < a + b$

$a - c < c < a + b$

$Q_{12}:$ In a triangle $ABC$ If $A<B<C$, then

$ac$

$a=b=c$

$a>b>c$

$a

$Q_{13}:$ Which of the following expresses the possible values of $p$, if $p$ is the perimeter of $RST $?

$3<p<22$

$19<p<22$

$19<p<38$

$22<p<38$

$Q_{14}:$ An isosceles triangle has sides of lengths $5, 11,$ and $x$. How many possible values of $x$ exist?

One

Two

Three

More than three

$Q_{15}:$ The distance between points $A$ and $D$ is $6$, and the distance between $D$ and $F$ is $4$. Which of the following is NOT a possible value for the distance between $F$ and $A $?

$3$

$4$

$7$

$11$

$Q_{16}:$ In the figure below, $r$ must be

$r=\sqrt{145}$

$r>\sqrt{145}$

$r<\sqrt{145}$

$1<r<17$

$Q_{17}:$ Two sides of a triangle have lengths $8$ and $4$. Find the largest possible integer lengths of the third side?

$4$

$6$

$11$

$12$

$Q_{18}:$ In triangle $ABC$, if $AB<AC$ and$m\angle C=30$, then $m\angle B$ must be

$m\angle B =30$

$m\angle B =60$

$m\angle B >30$

$m\angle B >60$

$Q_{19}:$ List the angles in order of smallest to largest.

$m\angle C< m\angle B<m\angle A$

$ m\angle B<m\angle C <m\angle A$

$ m\angle B<m\angle A <m\angle C$

$ m\angle A<m\angle C <m\angle B$

$Q_{20}:$ List the sides in order of longest to shortest.

$c<b<a$

$a<b<c$

$b<c<a$

$b<a<c$

$Q_{21}:$ List the sides in order of smallest to largest.

$t, k,y,r$

$t, k,r,y$

$k,t,y,r$

$t,y,k,r$

$Q_{22}:$ Which of the following is true?

$x<9$

$x<10$

$x<11$

$x<12$

$Q_{23}:$ Given the $2$ angles shown, determine which side is the longest side of the triangle?

$overline{TZ}$

$overline{TW}$

$overline{ZW}$

None

$Q_{24}:$ Given $\Delta ABC$ as shown. Which statement is NOT true?

$m\angle 1 = m\angle A + m\angle C$

$m\angle 1 > m\angle A$

$m\angle 1 + m\angle ABC =180$

$m\angle 1 < m\angle C$

$Q_{25}:$ What conclusions can you draw about $x $?

$1<x<13$

$5<x<13$

$0<x<29$

$0<x<10.3$

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