SAT · Math · Geometry and Trigonometry
Ángulos entre rectas paralelas, suma de ángulos de un triángulo y triángulos semejantes.
Un triángulo tiene ángulos de $50\circ$ y $60\circ$ → el tercero mide $180\circ-110\circ=70\circ$.
Intenta responder antes de abrir la solución. Las preguntas están en el idioma del examen.
In the figure, two lines intersect at point $O$. What is the value of $x$?
Respuesta correcta: C. $35$
The two labeled angles are opposite each other (vertical angles), so they are equal: $3x-10=2x+25$, which gives $x=35$.
$95$ is the measure of each angle, $3(35)-10$, not $x$. $33$ comes from treating the angles as adding up to $180\circ$. $7$ is $\frac{35}{5}$.
Each interior angle of a regular polygon measures $150\circ$. How many sides does the polygon have?
Respuesta numérica (escribes el número).
Respuesta: 12
Each exterior angle measures $180-150=30\circ$, and the exterior angles of any convex polygon add up to $360\circ$. So the polygon has $\frac{360}{30}=12$ sides.
Check: the interior angles of a $12$-sided polygon add up to $(12-2)\cdot 180=1{,}800\circ$, and $\frac{1{,}800}{12}=150$.
In the figure shown, point $D$ lies on $\overline{AB}$, point $E$ lies on $\overline{AC}$, and $\angle ADE$ is congruent to $\angle ACB$. If $AD=4$, $AE=6$, and $AC=12$, what is the length of $\overline{DB}$?
Respuesta numérica (escribes el número).
Respuesta: 14
Triangles $ADE$ and $ACB$ share angle $A$, and $\angle ADE\cong\angle ACB$, so the triangles are similar, with $A\leftrightarrow A$, $D\leftrightarrow C$, and $E\leftrightarrow B$. Corresponding sides are proportional: $\frac{AD}{AC}=\frac{AE}{AB}$, so $\frac{4}{12}=\frac{6}{AB}$ and $AB=18$. Then $DB=AB-AD=18-4=14$.
Careful: $\overline{DE}$ is not parallel to $\overline{BC}$, so $\frac{AD}{AB}=\frac{AE}{AC}$ (which would give $AB=8$ and $DB=4$) is not correct here.